Patent Publication Number: US-2010117908-A2

Title: Multi-metamaterial-antenna systems with directional couplers

Description:
PRIORITY CLAIMS AND RELATED APPLICATIONS  
      This document claims the benefits of the following four U.S. Provisional Patent Applications:  
      1. Ser. No. 61/016,392 entitled “Advanced Metamaterial Multi-Antenna Subsystems” and filed on Dec. 21, 2007;  
      2. Ser. No. 61/054,101 entitled “Metamaterial Antenna with Multiple Antenna Elements for Dual-Band Operations” and filed on May 16, 2008;  
      3. Ser. No. 61/098,730 entitled “Advanced Metamaterial Multi-Antenna System” and filed on Sep. 19, 2008; and  
      4. Ser. No. 61/098,731 entitled “Multi-Band Multi-Antenna System” and filed on Sep. 19, 2008.  
      The entire disclosures of the above applications are incorporated by reference as part of the disclosure of this document. 
    
    
     BACKGROUND  
      The propagation of electromagnetic waves in most materials obeys the right handed rule for the (E,H,β) vector fields, where E is the electrical field, β is the magnetic field, and β is the wave vector. The phase velocity direction is the same as the direction of the signal energy propagation (group velocity) and the refractive index is a positive number. Such materials are “right handed” (RH). Most natural materials are RH materials. Artificial materials can also be RH materials.  
      A metamaterial (MTM) has an artificial structure. When designed with a structural average unit cell size p much smaller than the wavelength of the electromagnetic energy guided by the metamaterial, the metamaterial can behave like a homogeneous medium to the guided electromagnetic energy. Unlike RH materials, a metamaterial can exhibit a negative refractive index with permittivity ∈ and permeability p being simultaneously negative, and the phase velocity direction is opposite to the direction of the signal energy propagation where the relative directions of the (E,H,β) vector fields follow the left handed rule. Metamaterials that support only a negative index of refraction with permittivity ∈ and permeability μ being simultaneously negative are pure “left handed” (LH) metamaterials.  
      Many metamaterials are mixtures of LH metamaterials and RH materials and thus are Composite Left and Right Handed (CRLH) metamaterials. A CRLH metamaterial can behave like a LH metamaterial at low frequencies and a RH material at high frequencies. Designs and properties of various CRLH metamaterials are described in, Caloz and Itoh, “Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications,” John Wiley &amp; Sons (2006). CRLH metamaterials and their applications in antennas are described by Tatsuo Itoh in “Invited paper: Prospects for Metamaterials,” Electronics Letters, Vol. 40, No. 16 (August, 2004).  
      CRLH metamaterials can be structured and engineered to exhibit electromagnetic properties that are tailored for specific applications and can be used in applications where it may be difficult, impractical or infeasible to use other materials. In addition, CRLH metamaterials may be used to develop new applications and to construct new devices that may not be possible with RH materials.  
     SUMMARY  
      Examples of apparatus and techniques for providing metamaterial (MTM) multi-antenna array systems with directional couplers are described for various applications. In one aspect, such a system includes two or more MTM antennas spaced from one another and each MTM antenna includes at least one unit cell which includes a series inductor, a shunt capacitor, a shunt inductor, and a series capacitor that are structured to form a composite right and left handed (CRLH) MTM structure. This system includes an MTM directional coupler comprising MTM transmission lines that are coupled to the MTM antennas and each MTM transmission line transmits a signal to or receives a signal from a respective MTM antenna. Each MTM transmission line includes a transmission line section, a shunt inductor, and a series capacitor that are structured to form a CRLH MTM structure and that are configured relative to an adjacent MTM transmission line coupled to an adjacent MTM antenna to reduce coupling between adjacent MTM antennas. In one implementation of this system, each MTM antenna is structured to exhibit two different resonance frequencies, each being a frequency different from a harmonic frequency of the other. In another implementation, this system includes a signal filter coupled to an MTM transmission line of the MTM directional coupler to transmit a selective frequency while blocking other frequencies.  
      In another aspect, an MTM multi-antenna array system for decoupling N number of signals between N number of antennas is provided to include an N-element metamaterial (MTM) antenna array; and an N-way directional coupler coupled to the N-element MTM antenna array. The N-way directional coupler has 2N ports.  
      These and other aspects and various implementations and their variations are described in detail in the attached drawings, the detailed description and the claims. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       FIG. 1  illustrates an example of a 1D CRLH MTM TL based on four unit cells.  
       FIG. 2  illustrates an equivalent circuit of the 1D CRLH MTM TL shown in  FIG. 1 .  
       FIG. 3  illustrates another representation of the equivalent circuit of the 1D CRLH MTM TL shown in  FIG. 1 .  
       FIG. 4A  illustrates a two-port network matrix representation for the 1D CRLH TL equivalent circuit shown in  FIG. 2 .  
       FIG. 4B  illustrates another two-port network matrix representation for the 1D CRLH TL equivalent circuit shown in  FIG. 3 .  
       FIG. 5  illustrates an example of a 1D CRLH MTM antenna based on four unit cells.  
       FIG. 6A  illustrates a two-port network matrix representation for the 1D CRLH antenna equivalent circuit analogous to the TL case shown in  FIG. 4A .  
       FIG. 6B  illustrates another two-port network matrix representation for the 1D CRLH antenna equivalent circuit analogous to the TL case shown in  FIG. 4B .  
       FIG. 7A  illustrates an example of a dispersion curve for the balanced case.  
       FIG. 7B  illustrates an example of a dispersion curve for the unbalanced case.  
       FIG. 8  illustrates an example of a 1D CRLH MTM TL with a truncated ground based on four unit cells.  
       FIG. 9  illustrates an equivalent circuit of the 1D CRLH MTM TL with the truncated ground shown in  FIG. 8 .  
       FIG. 10  illustrates an example of a 1D CRLH MTM antenna with a truncated ground based on four unit cells.  
       FIG. 11  illustrates another example of a 1D CRLH MTM TL with a truncated ground based on four unit cells.  
       FIG. 12  illustrates an equivalent circuit of the 1D CRLH MTM TL with the truncated ground shown in  FIG. 11 .  
       FIG. 13  illustrates a Multi-Antenna System comprising an N-element antenna array and an N-way directional coupler.  
       FIG. 14  illustrates an N-way directional coupler.  
       FIG. 15  illustrates an N-way metamaterial directional coupler.  
       FIG. 16  illustrates a configuration of the three-antenna system.  
       FIG. 17A  illustrates a structure of a three-element metamaterial antenna array: top view of top layer.  
       FIG. 17B  illustrates a structure of a three-element metamaterial antenna array: top view of bottom layer.  
       FIG. 18  illustrates a structure of a three-element metamaterial antenna array: 3-D view.  
       FIG. 19  illustrates simulated results of the three-element metamaterial antenna array shown in  FIGS. 17A, 17B , and  18 .  
       FIG. 20  illustrates a structure of the three-way directional coupler with six-ports: 3-D view.  
       FIG. 21  illustrates simulated results of the three-way directional coupler shown in  FIG. 20  for the input signal at P 1 .  
       FIG. 22  illustrates simulated results of the three-way directional coupler shown in  FIG. 20  for the input signal at P 2 .  
       FIG. 23A  illustrates a three-antenna system: top view.  
       FIG. 23B  illustrates a three-antenna system: bottom view.  
       FIG. 24  illustrates a structure of the three-antenna system: 3-D view.  
       FIG. 25  illustrates measured results of the three-antenna system shown in  FIG. 24 .  
       FIG. 26  illustrates measured radiation efficiencies for the three antennas in the three-antenna system shown in  FIG. 24 .  
       FIG. 27  illustrates a three-way MTM coupler.  
       FIG. 28  illustrates simulated results of the three-way MTM coupler shown in  FIG. 27  for the input signal at P 1 .  
       FIG. 29  illustrates simulated results of the three-way MTM coupler shown in  FIG. 27  for the input signal at P 2 .  
       FIG. 30  illustrates simulated results of the three-antenna system using three-way MTM coupler.  
       FIG. 31A  illustrates an example of a multi-antenna system configuration.  
       FIG. 31B  illustrates one implementation of the multi-antenna system configuration shown in  FIG. 31A .  
       FIGS. 32A-32D  illustrates an example of a multi-antenna system structure. A) 3-D view. B) Top view. C) Bottom view. D) Cross sectional view.  
       FIG. 33  illustrates the implementation of antenna array portion of the multi-antenna system structure shown in  FIG. 31 .  
       FIG. 34  illustrates an example of a microwave directional coupler that can be used in a multi-antenna system shown in  FIG. 31 .  
       FIG. 35  illustrates the return losses and isolation results of the metamaterial antenna array shown in  FIG. 33 .  
       FIG. 36  illustrates the return losses and isolation results of the multi-antenna system example shown in  FIG. 32 .  
       FIGS. 37A-37C  illustrates the radiation patterns of the multi-antenna system shown in  FIGS. 32A-32D . A) x-z plane. B) y-z plane. C) x-y plane.  
       FIGS. 38A-38B  illustrates A) Fabricated multi-antenna system. B) Measured return losses and isolation for multi-antenna system example shown in  FIGS. 32A-32D .  
       FIG. 39  illustrates the measured radiation efficiencies of multi-antenna system shown in  FIGS. 32A-32D  and metamaterial antenna array shown in  FIG. 33 .  
       FIGS. 40A-40D  illustrates an example of a multi-antenna system A) 3-D view. B) Top view. C) Bottom view. C) Cross sectional view.  
       FIGS. 41A-41C  illustrates various elements of an MTM coupler for the multi-antenna system shown in  FIGS. 40A-40D .  
       FIG. 42  illustrates simulation results of the return losses and isolation of the multi-antenna system shown in  FIGS. 40A-40D .  
       FIGS. 43A-43C  illustrates radiation patterns of the multi-antenna system shown in  FIG. 40A-40D  A) x-z plane. B) y-z plane. C) x-y plane.  
       FIGS. 44A-44C  illustrates A) Fabricated multi-antenna system shown in  FIGS. 40A-40D . B) Fabricated MTM coupler. C) Measured return losses and isolation for multi-antenna system Shown in  FIGS. 40A-40D .  
       FIG. 45  illustrates the measured radiation efficiencies of multi-antenna system shown in  FIGS. 40A-40D  and metamaterial antenna array shown in  FIG. 33 .  
       FIGS. 46A-46D  illustrates an example of a multi-antenna system structure. A) 3-D view. B) Top view. C) Bottom view. D) Cross sectional view.  
       FIGS. 47A-47C  illustrates various elements of the metamaterial antenna array with a metamaterial transmission line feed.  
       FIG. 48  illustrates an example of the MTM coupler for multi-antenna system shown in  FIGS. 46A-46D .  
       FIG. 49  illustrates the simulation results of the multi-antenna system shown in  FIGS. 46A-46D .  
       FIGS. 50A-50C  illustrates the radiation patterns of the multi-antenna system shown in  FIGS. 46A-46D . A) x-z plane. B) y-z plane. C) x-y plane.  
       FIGS. 51A-51D  illustrates a multi-antenna system structure. A) 3-D view. B) Top view. C) Bottom view. D) Cross sectional view.  
       FIGS. 52A-52C  illustrates a configuration of the multi-antenna system for USB application in detail.  
       FIG. 53  illustrates an simulation results of the metamaterial antenna array shown in  FIGS. 52A-52C  without the CPW MTM coupler.  
       FIG. 54  illustrates a simulation results of the multi-antenna system shown in  FIGS. 52A-52C .  
       FIGS. 55A-55C  illustrates the radiation patterns of the multi-antenna system shown in  FIGS. 52A-52C . A) x-z plane. B) y-z plane. C) x-y plane.  
       FIG. 56A-56B  illustrates the use of the multi-antenna systems for a time division duplex application.  
       FIG. 57A  illustrates a dualband multi-antenna system.  
       FIG. 57B  illustrates one implementation of the dualband multi-antenna system shown in  FIG. 57A .  
       FIGS. 58A-58C  illustrates individual layers of one implementation of dualband multi-antenna system.  
       FIG. 59  illustrates simulated results of metamaterial antenna array shown in  FIGS. 59A-59C .  
       FIGS. 60A-60B  illustrates A) microwave directional coupler. B) simulation results of microwave directional coupler.  
       FIG. 61  illustrates simulation results of the dualband multi-antenna system shown in  FIGS. 59A-59C .  
       FIG. 62  illustrates a dualband metamaterial antenna array.  
       FIGS. 63A-63B  illustrates the dualband metamaterial antenna array A) Top View of Top Layer. B) Top View of Bottom Layer.  
       FIGS. 64A-64B  illustrates simulation results of the dualband metamaterial antenna array shown in FIGS.  62 ,  63 A- 63 B.  
       FIGS. 65A-65B  illustrates A) a microwave directional coupler. B) simulation results of the microwave directional coupler.  
       FIGS. 66A-66B  illustrates A) a dualband multi-antenna system. B) simulation results of the dualband multi-antenna system.  
       FIGS. 67A-67B  illustrates simulation results of one example of a metamaterial antenna array.  
       FIGS. 68A-68B  illustrates an equivalent circuit model of a metamaterial transmission line which is implemented by cascading N unit cells periodically.  
       FIG. 69  illustrates an equivalent circuit model of MTM coupler.  
       FIG. 70  illustrates simulation results the MTM coupler.  
       FIG. 71  illustrates simulation results the dualband multi-antenna System using MTM coupler shown in  FIG. 69 .  
       FIGS. 72A-72E  illustrates a metamaterial antenna array. A) Layer1. B) Layer2. C) Layer3. D) Layer4. E) Four-Layer FR-4.  
       FIG. 73  illustrates a 3D view of the metamaterial Antenna Array shown in  FIGS. 72A-72E .  
       FIG. 74  illustrates measurement results of the metamaterial antenna array shown in  FIGS. 72A-72E  and  FIG. 73 .  
       FIGS. 75A-75E  illustrates a vertical directional coupler A) Layer1. B) Layer2. C) Layer3. D) Layer4. E) Four-Layer FR-4.  
       FIG. 76  illustrates simulation results of the vertical directional coupler shown in  FIGS. 75A-75E .  
       FIGS. 77A-77E  illustrates a dualband multi-antenna system using vertical directional coupler A) Layer1. B) Layer2. C) Layer3. D) Layer4. E) Four-Layer FR-4.  
       FIG. 78  illustrates measurement results of the dualband multi-antenna system shown in  FIGS. 77A-77E .  
       FIGS. 79A-79B  illustrates a MTM coupler with A) a LC network connecting in between two metamaterial transmission lines. B) a series capacitor and a series inductor connecting in between two metamaterial transmission lines.  
       FIGS. 80A-80C  illustrates multiple views of the small dualband multi-antenna system which have two metamaterial antennas and a MTM coupler in which A) represents layer  1 , B) represents layer  2 , and C) cross section view of layers  1  and  2  and substrate.  
       FIG. 81  illustrates the simulated return losses and coupling of the small dualband multi-antenna system shown in  FIGS. 80A-80C .  
       FIGS. 82A-82D  illustrates A) Generalized circuit model of a FW MTM coupler. B) Generalized circuit model of the FW MTM coupler with two parallel metamaterial transmission lines. C) Planar FW MTM coupler. D) Generalized circuit model of a asymmetric FW MTM coupler.  
       FIGS. 83A-83D  illustrates a vertical FW MTM coupler A) view of overlapping top layer and bottom layer. B) side view. C) top view of bottom layer. D) top view of top layer.  
       FIGS. 84A-84C  illustrates simulation results of the planar FW MTM coupler with C L1  variation.  
       FIGS. 85A-85C  illustrates simulation results of the planar FW MTM coupler with L m1  variation.  
       FIG. 86  illustrates simulation results of the vertical FW MTM coupler shown in  FIGS. 83A-83D .  
       FIGS. 87A-87B  illustrates a dualband multi-antenna system A) top view. B) 3D view.  
       FIGS. 88A-88C  illustrates a vertical FW MTM coupler A) top view of overlapping layer 1 , layer 2 , layer 3  and layer 4 . B) side view. C) more details of side view.  
       FIGS. 89A-89D  illustrates individual layers of vertical FW MTM directional coupler A) Layer 1. B) Layer 2. C) Layer 3. D) Layer 4.  
       FIG. 90  illustrates simulation results of the vertical FW MTM coupler.  
       FIGS. 91A-91C  illustrates a metamaterial antenna array A) top view of overlapping top layer and bottom layer. B) top view of top layer. C) top view of bottom layer.  
       FIG. 92  illustrates simulation results of the MTM antenna array shown in  FIGS. 91A-91C .  
       FIG. 93  illustrates simulation results of the dualband multi-antenna system shown in  FIGS. 87A-87B .  
       FIG. 94  illustrates a multi-band multi-antenna system.  
       FIGS. 95A-95F  illustrates metamaterial WiFi and WiMax antenna array with A) top view of substrate I. B) bottom view substrate I. C) top view of substrate II. D) bottom view of substrate II. E) top view of substrate III. F) bottom view of substrate III.  
       FIG. 96  illustrates a 3D view of the metamaterial WiFi and WiMax antenna array.  
       FIG. 97  illustrates simulated results of the metamaterial WiFi and WiMax antenna array shown in  FIGS. 95A-95F  and  FIG. 96 .  
       FIG. 98  illustrates a microwave coupled line coupler.  
       FIG. 99  illustrates simulated results of the microwave coupled line coupler shown in  FIG. 98 .  
       FIG. 100  illustrates simulated results of the multi-band multi-antenna system with the microwave coupled line coupler.  
       FIG. 101  illustrates a MTM coupler.  
       FIG. 102  illustrates simulated results of the MTM coupler shown in  FIG. 101 .  
       FIG. 103  illustrates simulated results of the multi-band multi-antenna system with the MTM coupler.  
       FIG. 104  illustrates a multi-band multi-antenna system with bandpass filters.  
       FIGS. 105A-105B  illustrates A) a Chebyshev WiFi bandpass filter (prototype). B) a Chebyshev WiMax bandpass filter (prototype).  
       FIG. 106  illustrates simulated results of the Chebyshev WiFi and WiMax bandpass filters shown in  FIGS. 105A-105B .  
       FIG. 107  illustrates simulated results of the multi-band multi-antenna system shown in  FIG. 104 .  
       FIG. 108  illustrates a multi-band multi-antenna system with a directional coupler and a bandpass filters.  
       FIG. 109  illustrates simulated results of the multi-band multi-antenna system with microwave coupled line coupler and bandpass filter.  
       FIG. 110  illustrates simulated results of the multi-band multi-antenna system with metamaterial directional coupler and bandpass filters. 
    
    
      In the appended figures, similar components and/or features may have the same reference numeral. Further, various components of the same type may be distinguished by following the reference numeral by a dash and a second label that distinguishes among the similar components. If only the first reference numeral is used in the specification, the description is applicable to any one of the similar components having the same first reference numeral.  
     DETAILED DESCRIPTION  
      Metamaterial (MTM) structures can be used to construct antennas and other electrical components and devices, allowing for a wide range of technology advancements such as size reduction and performance improvements. The MTM antenna structures can be fabricated on various circuit platforms, for example, a conventional FR-4 Printed Circuit Board (PCB) or a Flexible Printed Circuit (FPC) board. Examples of other fabrication techniques include thin film fabrication technique, system on chip (SOC) technique, low temperature co-fired ceramic (LTCC) technique, and monolithic microwave integrated circuit (MMIC) technique.  
      Exemplary MTM antenna structures are described in U.S. patent application Ser. No. 11/741,674 entitled “Antennas, Devices, and Systems Based on Metamaterial Structures,” filed on Apr. 27, 2007, and U.S. patent application Ser. No. 11/844,982 entitled “Antennas Based on Metamaterial Structures,” filed on Aug. 24, 2007, which are hereby incorporated by reference as part of the disclosure of this document.  
      An MTM antenna or MTM transmission line (TL) is a MTM structure with one or more MTM unit cells. The equivalent circuit for each MTM unit cell includes a right-handed series inductance (LR), a right-handed shunt capacitance (CR), a left-handed series capacitance (CL), and a left-handed shunt inductance (LL). LL and CL are structured and connected to provide the left-handed properties to the unit cell. This type of CRLH TLs or antennas can be implemented by using distributed circuit elements, lumped circuit elements or a combination of both. Each unit cell is smaller than ˜λ/4 where λ is the wavelength of the electromagnetic signal that is transmitted in the CRLH TL or antenna.  
      A pure LH metamaterial follows the left-hand rule for the vector trio (E,H,β), and the phase velocity direction is opposite to the signal energy propagation. Both the permittivity ∈ and permeability μ of the LH material are negative. A CRLH metamaterial can exhibit both left-hand and right-hand electromagnetic modes of propagation depending on the regime or frequency of operation. Under certain circumstances, a CRLH metamaterial can exhibit a non-zero group velocity when the wavevector of a signal is zero. This situation occurs when both left-hand and right-hand modes are balanced. In an unbalanced mode, there is a bandgap in which electromagnetic wave propagation is forbidden. In the balanced case, the dispersion curve does not show any discontinuity at the transition point of the propagation constant β(ω o )=0 between the left- and right-hand modes, where the guided wavelength is infinite, i.e., λ g =2π/|β|→∞, while the group velocity is positive:  
               v   g     =       ⅆ   ω       ⅆ   β                β   =   0       &gt;   0.       
 
 This state corresponds to the zeroth order mode m=0 in a TL implementation in the LH region. The CRHL structure supports a fine spectrum of low frequencies with the dispersion relation that follows the negative β parabolic region. This allows a physically small device to be built that is electromagnetically large with unique capabilities in manipulating and controlling near-field radiation patterns. When this TL is used as a Zeroth Order Resonator (ZOR), it allows a constant amplitude and phase resonance across the entire resonator. The ZOR mode can be used to build MTM-based power combiners and splitters or dividers, directional couplers, matching networks, and leaky wave antennas. 
 
      In the case of RH TL resonators, the resonance frequency corresponds to electrical lengths θ m =β m l=mπ (m=1, 2, 3 . . . ), where l is the length of the TL. The TL length should be long to reach low and wider spectrum of resonant frequencies. The operating frequencies of a pure LH material are at low frequencies. A CRLH MTM structure is very different from an RH or LH material and can be used to reach both high and low spectral regions of the RF spectral ranges. In the CRLH case θ m =β m l=mπ, where 1 is the length of the CRLH TL and the parameter m=0, ±1, ±2, ±3 . . . ±∞.  
       FIG. 1  illustrates an example of a 1D CRLH MTM TL based on four unit cells. One unit cell includes a cell patch and a via, and is a minimum unit that repeats itself to build the MTM structure. The four cell patches are placed on a substrate with respective centered vias connected to the ground plane.  
       FIG. 2  shows an equivalent network circuit of the 1D CRLH MTM TL in  FIG. 1 . The ZLin′ and ZLout′ correspond to the TL input load impedance and TL output load impedance, respectively, and are due to the TL coupling at each end. This is an example of a printed two-layer structure. LR is due to the cell patch on the dielectric substrate, and CR is due to the dielectric substrate being sandwiched between the cell patch and the ground plane. CL is due to the presence of two adjacent cell patches, and the via induces LL.  
      Each individual unit cell can have two resonances ω SE  and ω SH  corresponding to the series (SE) impedance Z and shunt (SH) admittance Y. In  FIG. 2 , the Z/2 block includes a series combination of LR/2 and 2CL, and the Y block includes a parallel combination of LL and CR. The relationships among these parameters are expressed as follows:  
                   ω   SH     =     1       LL   ⁢           ⁢   CR           ;       ω   SE     =     1       LR   ⁢           ⁢   CL           ;       ω   R     =     1       LR   ⁢           ⁢   CR           ;     ⁢     
     ⁢         ω   L     =       1       LL   ⁢           ⁢   CL         ⁢           ⁢   where       ,          ⁢     Z   =         j   ⁢           ⁢   ω   ⁢           ⁢   LR     +       1     j   ⁢           ⁢   ω   ⁢           ⁢   CL       ⁢           ⁢   and   ⁢           ⁢   Y       =       j   ⁢           ⁢   ω   ⁢           ⁢   CR     +     1     j   ⁢           ⁢   ω   ⁢           ⁢   LL                       Eq   .           ⁢     (   1   )               
 
      The two unit cells at the input/output edges in  FIG. 1  do not include CL, since CL represents the capacitance between two adjacent cell patches and is missing at these input/output edges. The absence of the CL portion at the edge unit cells prevents ω SE  frequency from resonating. Therefore, only ω SH  appears as an m=0 resonance frequency.  
      To simplify the computational analysis, a portion of the ZLin′ and ZLout′ series capacitor is included to compensate for the missing CL portion, and the remaining input and output load impedances are denoted as ZLin and ZLout, respectively, as seen in  FIG. 3 . Under this condition, all unit cells have identical parameters as represented by two series Z/2 blocks and one shunt Y block in  FIG. 3 , where the Z/2 block includes a series combination of LR/2 and 2CL, and the Y block includes a parallel combination of LL and CR.  
       FIG. 4A  and  FIG. 4B  illustrate a two-port network matrix representation for TL circuits without the load impedances as shown in  FIG. 2  and  FIG. 3 , respectively,  
       FIG. 5  illustrates an example of a 1D CRLH MTM antenna based on four unit cells.  FIG. 6A  shows a two-port network matrix representation for the antenna circuit in  FIG. 5 .  FIG. 6B  shows a two-port network matrix representation for the antenna circuit in  FIG. 5  with the modification at the edges to account for the missing CL portion to have all the unit cells identical.  FIGS. 6A and 6B  are analogous to the TL circuits shown in  FIGS. 4A and 4B , respectively.  
      In matrix notations,  FIG. 4B  represents the relationship given as below:  
               (         Vin           Iin         )     =       (         AN       BN           CN       AN         )     ⁢     (         Vout           Iout         )               Eq   .           ⁢     (   2   )               
 
 where AN=DN because the CRLH MTM TL circuit in  FIG. 3  is symmetric when viewed from Vin and Vout ends. 
 
      In  FIGS. 6A and 6B , the parameters GR′ and GR represent a radiation resistance, and the parameters ZT′ and ZT represent a termination impedance. Each of ZT′, ZLin′ and ZLout′ includes a contribution from the additional 2CL as expressed below:  
                 ZLin   ′     =     ZLin   +     2     j   ⁢           ⁢   ω   ⁢           ⁢   CL           ,     
     ⁢       ZLout   ′     =     ZLout   +     2     j   ⁢           ⁢   ω   ⁢           ⁢   CL           ,     
     ⁢       ZT   ′     =     ZT   +     2     j   ⁢           ⁢   ω   ⁢           ⁢   CL                   Eq   .           ⁢     (   3   )               
 
      Since the radiation resistance GR or GR′ can be derived by either building or simulating the antenna, it may be difficult to optimize the antenna design. Therefore, it is preferable to adopt the TL approach and then simulate its corresponding antennas with various terminations ZT. The relationships in Eq. (1) are valid for the circuit in  FIG. 2  with the modified values AN′, BN′, and CN′, which reflect the missing CL portion at the two edges.  
      The frequency bands can be determined from the dispersion equation derived by letting the N CRLH cell structure resonate with nπ propagation phase length, where n=0, ±1, ±2, . . . ±N. Here, each of the N CRLH cells is represented by Z and Y in Eq. (1), which is different from the structure shown in  FIG. 2 , where CL is missing from end cells. Therefore, one might expect that the resonances associated with these two structures are different. However, extensive calculations show that all resonances are the same except for n=0, where both ω SE  and ω SH  resonate in the structure in  FIG. 3 , and only ω SH  resonates in the structure in  FIG. 2 . The positive phase offsets (n&gt;0) correspond to RH region resonances and the negative values (n&lt;0) are associated with LH region resonances.  
      The dispersion relation of N identical CRLH cells with the Z and Y parameters is given below:  
             {                 N   ⁢           ⁢   β   ⁢           ⁢   p     =       cos     -   1       ⁡     (     A   N     )         ,       ⇒            A   N          ≤   1     ⇒     0   ≤   χ       =       -   ZY     ≤     4   ⁢     ∀   N                                 where   ⁢           ⁢     A             ⁢   N         =     1   ⁢           ⁢   at   ⁢           ⁢   even   ⁢           ⁢   resonances                      n        =       2   ⁢           ⁢   m     ∈     {           0   ,           ⁢   2   ,           ⁢   4   ,           ⁢     …   ⁢           ⁢   2   ×                 Int   (           ⁢       N   ⁢           -           ⁢   1               ⁢   2       )           }               ⁢                                 and   ⁢           ⁢     A             ⁢   N         =       -   1     ⁢           ⁢   at   ⁢           ⁢   odd   ⁢           ⁢   resonances                         ⁢            n        =         2   ⁢           ⁢   m     +   1     ∈     {     1   ,   3   ,     …   ⁢           ⁢     (       2   ×     Int   (           ⁢     N             ⁢   2       )       ⁢           -           ⁢   1     )         }         ,             ⁢                                 Eq   .           ⁢     (   4   )               
 
 where Z and Y are given in Eq. (1), AN is derived from the linear cascade of N identical CRLH unit cells as in  FIG. 3 , and p is the cell size. Odd n=(2 m+1) and even n=2 m resonances are associated with AN=−1 and AN=1, respectively. For AN′ in  FIG. 4A  and  FIG. 6A , the n=0 mode resonates at ω 0 =ω SH  only and not at both ω SE  and ω SH  due to the absence of CL at the end cells, regardless of the number of cells. Higher-order frequencies are given by the following equations for the different values of χ specified in Table 1:  
                 For   ⁢           ⁢   n     &gt;   0     ,       ω     ±   n     2     =             ω             ⁢   SH               ⁢   2       ⁢           +           ⁢     ω             ⁢   SE               ⁢   2       ⁢           +           ⁢     χ   ⁢           ⁢     ω             ⁢   R               ⁢   2           2     ⁢             ±                 (         ω   SH   2     ⁢           +           ⁢     ω   SE   2     ⁢           +           ⁢     χ   ⁢           ⁢     ω   R   2         2     )     2     -                 ω   SH   2     ⁢           ⁢     ω   SE   2                           Eq   ⁢           .           ⁢     (   5   )               
 
      Table 1 provides χ values for N=1, 2, 3, and 4. It should be noted that the higher-order resonances |n|&gt;0 are the same regardless if the full CL is present at the edge cells ( FIG. 3 ) or absent ( FIG. 2 ). Furthermore, resonances close to n=0 have small χ values (near χ lower bound 0), whereas higher-order resonances tend to reach χ upper bound 4 as stated in Eq. (4).  
               TABLE 1                          Resonances for N = 1, 2, 3 and 4 cells                                 N\Modes   |n| = 0   |n| = 1   |n| = 2   |n| = 3               N = 1   χ (1,0)  = 0; ω 0  = ω SH                     N = 2   χ (2,0)  = 0; ω 0  = ω SH     χ (2,1)  = 2       N = 3   χ (3,0)  = 0; ω 0  = ω SH     χ (3,1)  = 1   χ (3,2)  = 3       N = 4   χ (4,0)  = 0; ω 0  = ω SH     χ (4,1)  = 2 − {square root over (2)}   χ (4,2)  = 2                  
 
      The dispersion curve β as a function of frequency ω is illustrated in  FIGS. 7A and 7B  for the ω SE =ω SH  (balanced, i.e., LR CL=LL CR) and ω SE ≠ω SH  (unbalanced) cases, respectively. In the latter case, there is a frequency gap between min(ω SE , ω SH ) and max(ω SE , ω SH ). The limiting frequencies ω min  and ω max  values are given by the same resonance equations in Eq. (5) with χ reaching its upper bound χ=4 as stated in the following equations:  
                 ω   min   2     =           ω   SH   2     +     ω   SE   2     +     4   ⁢           ⁢     ω   R   2         2     -                 (         ω   SH   2     +     ω   SE   2     +     4   ⁢           ⁢     ω   R   2         2     )     2     -                 ω   SH   2     ⁢     ω   SE   2                   ⁢     
     ⁢     ω   max   2     =           ω   SH   2     +     ω   SE   2     +     4   ⁢           ⁢     ω   R   2         2     +                 (         ω   SH   2     +     ω   SE   2     +     4   ⁢           ⁢     ω   R   2         2     )     2     -                 ω   SH   2     ⁢     ω   SE   2                         (   6   )             
 
      In addition,  FIGS. 7A and 7B  provide examples of the resonance position along the dispersion curves. In the RH region (n&gt;0) the structure size 1=Np, where p is the cell size, increases with decreasing frequency. In contrast, in the LH region, lower frequencies are reached with smaller values of Np, hence size reduction. The dispersion curves provide some indication of the bandwidth around these resonances. For instance, LH resonances have the narrow bandwidth because the dispersion curves are almost flat. In the RH region, the bandwidth is wider because the dispersion curves are steeper. Thus, the first condition to obtain broadbands, 1 st  BB condition, can be expressed as follows:  
             COND   ⁢           ⁢   1   ⁢     :                                       1   st     ⁢           ⁢   BB   ⁢           ⁢   condition   ⁢              ⅆ   β       ⅆ   ω            res       =       ⁢              -         ⅆ     (   AN   )         ⅆ   ω           (     1   -     AN   2       )                res     ⁢     &lt;&lt;   1     ⁢             ⁢             ⁢   near   ⁢           ⁢   ω                 =       ⁢     ω   res                   =       ⁢     ω             ⁢   0         ,     ω     ±   1       ,         ω     ±   2       ⁢           ⁢   …     ⇒            ⅆ   β       ⅆ   ω                          =       ⁢                  ⅆ   χ       ⅆ   ω         2   ⁢           ⁢   p   ⁢       χ   ⁡     (     1   -     χ   4       )                  res     ⁢     &lt;&lt;   1     ⁢           ⁢   with   ⁢           ⁢   p                     =       ⁢     cell   ⁢             ⁢             ⁢   size   ⁢           ⁢   and   ⁢           ⁢       ⅆ   χ       ⅆ   ω                res               =       ⁢         2   ⁢           ⁢     ω     ±   n           ω   R   2       ⁢     (     1   -         ω   SE   2     ⁢     ω   SH   2         ω     ±   n     4         )                     Eq   .           ⁢     (   7   )               
 
 where χ is given in Eq. (4) and ω R  is defined in Eq. (1). The dispersion relation in Eq. (4) indicates that resonances occur when |AN|=1, which leads to a zero denominator in the 1 st  BB condition (COND1) of Eq. (7). As a reminder, AN is the first transmission matrix entry of the N identical unit cells ( FIG. 4B  and  FIG. 6B ). The calculation shows that COND1 is indeed independent of N and given by the second equation in Eq. (7). It is the values of the numerator and χ at resonances, which are shown in Table 1, that define the slopes of the dispersion curves, and hence possible bandwidths. Targeted structures are at most Np=λ/40 in size with the bandwidth exceeding 4%. For structures with small cell sizes p, Eq. (7) indicates that high ω R  values satisfy COND1, i.e., low CR and LR values, since for n&lt;0 resonances occur at χ values near 4 in Table 1, in other terms (1−χ/4→0). 
 
      As previously indicated, once the dispersion curve slopes have steep values, then the next step is to identify suitable matching. Ideal matching impedances have fixed values and may not require large matching network footprints. Here, the word “matching impedance” refers to a feed line and termination in the case of a single side feed such as in antennas. To analyze an input/output matching network, Zin and Zout can be computed for the TL circuit in  FIG. 4B . Since the network in  FIG. 3  is symmetric, it is straightforward to demonstrate that Zin=Zout. It can be demonstrated that Zin is independent of N as indicated in the equation below:  
                 Zin   2     =       BN   CN     =         B   ⁢           ⁢   1       C   ⁢           ⁢   1       =       Z   Y     ⁢     (     1   -     χ   4       )             ,           Eq   .           ⁢     (   8   )               
 
 which has only positive real values. One reason that B1/C1 is greater than zero is due to the condition of |AN|≦L in Eq. (4), which leads to the following impedance condition: 
 
0 ≦−ZY=χ≦ 4 .  
 
 The 2 nd  broadband (BB) condition is for Zin to slightly vary with frequency near resonances in order to maintain constant matching. Remember that the real input impedance Zin′ includes a contribution from the CL series capacitance as stated in Eq. (3). The 2 nd  BB condition is given below:  
             COND   ⁢           ⁢   2   ⁢     :                                       2   ed     ⁢           ⁢   BB   ⁢           ⁢   condition   ⁢     :     ⁢           ⁢   near   ⁢           ⁢   resonances     ,       ⅆ   Zin       ⅆ   ω                near   ⁢           ⁢   res       ⁢     &lt;&lt;   1             Eq   .           ⁢     (   9   )               
 
      Different from the transmission line example in  FIG. 2  and  FIG. 3 , antenna designs have an open-ended side with an infinite impedance which poorly matches the structure edge impedance. The capacitance termination is given by the equation below:  
                 Z   T     =     AN   CN       ,           Eq   .           ⁢     (   10   )               
 
 which depends on N and is purely imaginary. Since LH resonances are typically narrower than RH resonances, selected matching values are closer to the ones derived in the n&lt;0 region than the n&gt;0 region. 
 
      To increase the bandwidth of LH resonances, the shunt capacitor CR should be reduced. This reduction can lead to higher ω R  values of steeper dispersion curves as explained in Eq. (7). There are various methods of decreasing CR, including but not limited to: 1) increasing substrate thickness, 2) reducing the cell patch area, 3) reducing the ground area under the top cell patch, resulting in a “truncated ground,” or combinations of the above techniques.  
      The structures in  FIGS. 1 and 5  use a conductive layer to cover the entire bottom surface of the substrate as the full ground electrode. A truncated ground electrode that has been patterned to expose one or more portions of the substrate surface can be used to reduce the area of the ground electrode to less than that of the full substrate surface. This can increase the resonant bandwidth and tune the resonant frequency. Two examples of a truncated ground structure are discussed with reference to  FIGS. 8 and 11 , where the amount of the ground electrode in the area in the footprint of a cell patch on the ground electrode side of the substrate has been reduced, and a remaining strip line (via line) is used to connect the via of the cell patch to a main ground electrode outside the footprint of the cell patch. This truncated ground approach may be implemented in various configurations to achieve broadband resonances.  
       FIG. 8  illustrates one example of a truncated ground electrode for a four-cell transmission line where the ground has a dimension that is less than the cell patch along one direction underneath the cell patch. The ground conductive layer includes a via line that is connected to the vias and passes through underneath the cell patches. The via line has a width that is less than a dimension of the cell path of each unit cell. The use of a truncated ground may be a preferred choice over other methods in implementations of commercial devices where the substrate thickness cannot be increased or the cell patch area cannot be reduced because of the associated decrease in antenna efficiencies. When the ground is truncated, another inductor Lp ( FIG. 9 ) is introduced by the metallization strip (via line) that connects the vias to the main ground as illustrated in  FIG. 8 .  FIG. 10  shows a four-cell antenna counterpart with the truncated ground analogous to the TL structure in  FIG. 8 .  
       FIG. 11  illustrates another example of a truncated ground structure. In this example, the ground conductive layer includes via lines and a main ground that is formed outside the footprint of the cell patches. Each via line is connected to the main ground at a first distal end and is connected to the via at a second distal end. The via line has a width that is less than a dimension of the cell path of each unit cell.  
      The equations for the truncated ground structure can be derived. In the truncated ground examples, CR becomes very small, and the resonances follow the same equations as in Eqs. (1), (5) and (6) and Table 1 as explained below:  
      Approach 1 ( FIGS. 8 and 9 )  
      Resonances: same as in Eqs. (1), (5) and (6) and Table 1 after replacing LR by LR+Lp.  
      Furthermore, for |n|≠0, each mode has two resonances corresponding to  
     
         
          (1) ω±n for LR being replaced by LR+Lp  
          (2) ω±n for LR being replaced by LR+Lp/N where N is the number of cells 
 
 The impedance equation becomes:  
                 Zin   2     =       BN   CN     =         B   ⁢           ⁢   1       C   ⁢           ⁢   1       =       Z   Y     ⁢     (     1   -       χ   +     χ   P       4       )     ⁢       (     1   -   χ   -     χ   P       )       (     1   -   χ   -       χ   P     /   N       )               ,     
     ⁢       where   ⁢           ⁢   χ     =         -   YZ     ⁢           ⁢   and   ⁢           ⁢   χ     =     -     YZ   p           ,     
     ⁢       where   ⁢           ⁢   Zp     =     j   ⁢           ⁢   ω   ⁢           ⁢   Lp   ⁢           ⁢   and   ⁢           ⁢   Z       ,     Y   ⁢           ⁢   are   ⁢           ⁢   defined   ⁢           ⁢   in   ⁢           ⁢     Eq   .           ⁢     (   2   )     .               Eq   .           ⁢     (   11   )               
 
 From the impedance equation in Eq. (11), it can be seen that the two resonances ω and ω′ have low and high impedances, respectively. Thus, it is easy to tune near the ω resonance in most cases. 
 
 Approach 2 ( FIGS. 11 and 12 ) 
 
 Resonances: same as in Eqs. (1), (5), and (6) and Table 1 after replacing LL by LL+Lp. 
 
 In the second approach, the combined shunt inductor (LL+Lp) increases while the shunt capacitor CR decreases, which leads to lower LH frequencies. 
 
       
    
      Modern wireless communication systems use multiple antennas to improve the performance namely, capacity, reliability or coverage. Receive diversity, beam-switching and Multiple-Input-Multiple-Output (MIMO) systems are a few examples of communication systems that can benefit from such advanced multi-antenna systems. Multiple Input Multiple Output (MIMO) is the most promising and challenging wireless transmission technology to improve the capacity of wireless systems. MIMO techniques combine signals from multiple antennas to exploit the multipath in wireless channel and enable higher capacity, better coverage and increased reliability. The key requirement to realize the benefits of multi-antenna systems is to send/receive multiple signals with minimum correlation at the air interface. However, the antenna element spacing needed to minimize the coupling between antennas is 0.5λ 0  where λ 0  is the free space wavelength. This requirement can hinder practical application of MIMO designs based on some other antenna designs. Furthermore most wireless communication standards require operation over multiple bands for world-wide coverage or due to frequency allocation.  
      Consumer devices like cell phones, Smart phones and client cards continue to shrink in size and the room available for antennas is getting smaller. There are various technical challenges associated with realizing the multiband multi-antenna system in such practical applications. The first challenge is to design a single input multiband antenna in a compact size without compromising radiation efficiency. The second and more challenging issue is to minimize the interaction between the antennas that are placed in very close proximity across all operating bands. The minimum coupling between two closely coupled antennas can be achieved by placing antenna elements half-wavelength away from each other. However, this is not practical in commercial products because of the limited space. If the interaction between antennas is not minimized, the MIMO benefits cannot be obtained.  
      One of the approaches to improving the isolation for the closely coupled antenna is to integrate microwave directional coupler and antennas into the multi-antenna system. However, the size of conventional microwave coupler prevents it from the practical usage. In addition, the printed circuit board (PCB) fabrication process for the microwave circuit will make the conventional microwave coupler difficult to achieve more than −8 dB coupling. This restriction limits the spacing of the antenna array used in the multi-system, such as MIMO, to at most one sixth of the wavelength. The available area in many wireless devices is generally restricted to a small spacing between two adjacent antennas, e.g., 0.1λ 0 ˜0.25λ 0  or less, where λ 0  is the free space wavelength. In addition to the single band couple, dualband or multi-band couplers can also be designed.  
      Metamaterial technology has the advantage of 1) reducing the circuit size while providing equivalent or better performance for antenna and 2) improving isolation in antenna arrays by confining near-fields in a small area. The dispersion engineering used in MTM technology can control the propagation constant and the characteristic impedance of the transmission line so that the physical size of circuit may be independent of the operational frequency and can be significantly reduced to fit in a small area. The metamaterial technology can solve both the challenges (1 and 2) described above. A metamaterial antenna can support multiple frequencies in a small, low-profile and low cost form. Using metamaterial technology, the coupler circuit physical size is independent of the operational frequency and can be significantly reduced to fit in a small area.  
      The technical features in this document can be used to decouple N coupled antenna elements using an N-way directional coupler. The N-element antenna array can be implemented by using either conventional antennas with right-handed material properties or metamaterial antennas such as CRLH MTM antennas. The N-way directional coupler can be implemented by using conventional transmission lines with right-handed material properties or metamaterial transmission lines. One of advantages for using the metamaterial technology is that the physical size of circuits can be significantly reduced to fit in modern communication system. A metamaterial coupler may also be configured to provide up to 0 dB coupling which cannot be done by using conventional directional coupler. Certain information on features described in this document can also be found in Caloz and Itoh, “Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications,” John Wiley &amp; Sons, 2006; and Caloz et al., “Generalized Coupled-Mode Approach of Metamaterial Coupled-Line Couplers: Coupling Theory, Phenomenological Explanation, and Experimental Demonstration, IEEE Transactions on Microwave Theory and Techniques, Vol. 55, No. 5, May 2007.  
      Examples of multiband antenna systems in this document combine a multiband metamaterial antenna array (Metarray™) and either a microwave directional coupler or metamaterial directional coupler (MTM coupler) in a planar form to reduce the coupling arising from the proximity effects of antenna array elements. All the components are jointly optimized to minimize coupling and maximize orthogonality of radiation patterns at multiple frequencies. Examples of multi-antenna systems using metamaterial structures are described below to illustrate various antenna features and antenna system features that can increase spectral efficiency and channel capacity. The metamaterial structures can be configured to increase isolation between different input ports and restore orthogonality between multi-path signals in the analog domain. The systems described in this document can include multiple antennas and a network of couplers where at least one antenna or coupler is based on metamaterial technology.  
      The metamaterial antenna systems described in this document can also be configured to enable applications that may be impractical or technically difficult to implement based on conventional RF antenna designs using right-handed materials. For example, metamaterial antenna systems described in this document can be designed to achieve high isolation to enable full duplex communication in time division duplex systems. Such operations to date have been considered impractical by using conventional RF antenna designs due to the high coupling between transmitted and received signals.  
      For example, one approach presented in this document for enhancing the isolation of coupled antenna elements is to incorporate a directional coupler in the antenna system. The directional coupler can eliminate the unwanted coupling signal from the adjacent antenna elements. This can be done by optimizing the coupling magnitude and phase of the directional coupler based on the coupling and phase between the antenna elements. The challenge here is to satisfy the magnitude and phase requirements at multiple frequencies in order to design a multiband multi-antenna system. This document describes various different approaches to realize such multiband multi-antenna systems.  
      A multi-antenna system may be structured to include closely spaced antenna elements and make each antenna support a different frequency band. The isolation between the two antenna elements are desirable when such a multi-antenna system is used in various applications. For example, access devices such as home gateways may require support for WiFi and WiMax technologies on the same board to create a transition from WLAN to WWAN. Integrating WiFi and WiMax technologies can create significant implementation challenges due to cross talk and isolation issues between WiFi and WiMax frequency bands. Because WiFi and WiMax operate independently, isolation can be an important factor to prevent WiMax radio transmissions from blocking or interfering with WiFi radio transmission, which may be receiving or transmitting data. One possible solution for addressing isolation issues is the use of a filter to suppress the interference between the two closely spaced frequency bands. The filter, however, typically requires a design that is characterized by a flat response in a passband frequency range and a sharp rejection just outside the passband frequency range. For example, to achieve adequate isolation in the WiFi and WiMax frequency bands, the filter should have a passband frequency range of about 2.4 GHz to 2.48 GHz and a rejection that is better than 30 dB at 2.5 GHz and higher. State of the art surface acoustic wave (SAW) and bulk acoustic wave (BAW) filters can achieve the rejection performance but at an increased expense in cost and insertion loss (typically 2-3 dB). Because these filters are placed after the power amplifier or in the receiver path before low noise amplifier (LNA), they can create significant loss in the link budget. In mass production, to meet these sharp transition requirements, high tolerance components need to be used to maintain desired production yields. This increases the manufacturing cost of these filters.  
      In this regard, a combination of a coupler and filters with slow roll-off in the filter response may be used to meet the antenna rejection requirements without compromising the insertion loss. One reason for this can be attributed to the opposite transfer characteristics of the coupler and the filter. Typically, the coupler can offer good isolation between two ports over a narrow bandwidth. By positioning the coupler isolation band between the two closely-spaced frequency bands, lower filter rejection requirements can be achieved. In a conventional method, typical solutions generally involve the use of a large coupler and filter components and thus may be impractical to implement due to size constraints in certain applications. The metamaterial technology can provide an advantage of reducing circuit size while maintaining or improving performances.  
      The RF structures and antenna designs in this document can be implemented by using printed circuit boards, such as FR-4 printed circuit boards. Examples of other fabrication techniques include thin film fabrication technique, system on chip (SOC) technique, low temperature co-fired ceramic (LTCC) technique, and monolithic microwave integrated circuit (MMIC) technique.  
      Various features described in this document include: design rules for the microwave directional coupler and metamaterial directional coupler based on different single-band or multi-band antenna arrays; design of a multi-antenna system including two metamaterial antenna elements and a conventional microwave directional coupler; designs and implementations of a multi-antenna system which includes two metamaterial antenna elements and a metamaterial directional coupler; metamaterial couplers with backward wave (BW) or forward wave (FW) coupling; and introduction of additional discrete or printed components to increase the mutual capacitive or inductive coupling between the two lines. Various implementation examples are provided in this document, including examples of using planar and vertical directional couplers and examples of using coupled microstrip or coplanar waveguide (CPW).  
      The above design approaches can be applied to other types of directional couplers such as coupled lines fully embedded inside dielectric substrates.  
      I. Multi-Antenna Array Systems with Directional Couplers  
      A multi-antenna system include two or more antennas coupled in close proximity in a device.  FIG. 13  illustrates a multi-antenna system  1300  comprising an N-element antenna array  1301 . Such a system can be designed to have high coupling between adjacent antennas such as Ant 1  and Ant 2 , (Ant 2  and Ant 3 ), and AntN- 1  and AntN as shown. In such a system, coupling between two non-adjacent antennas, that are separated by one or more antennas and thus are not immediate adjacent to each other, can be much smaller than coupling between adjacent antennas and, thus, has less impact to the system performance then coupling between adjacent antennas.  
      In  FIG. 13 , an N-way directional coupler  1315  is introduced to decouple the N antenna elements forming an N-element antenna array  1301 . The N-way directional coupler  1315  can be structured to include input ports  1320  (P 1 , P 2 , . . . , PN) and output ports  1310  (PN+1, PN+2, . . . , P2N) which are respectively connected to ports  1305  (P 1 ′, P 2 ′, . . . , PN′) of the N-element antenna array  1301 . Based on the coupling behavior for the N-element antenna array  1301 , the N-way directional coupler  1315  should be designed so that the coupled signals between Pm and Pm+1 where m=1′, 2′, 2N−1′ are decoupled. The N-way directional coupler  1315  can be implemented by using either a metamaterial technology or non-metamaterial approach.  
       FIG. 14  shows an example of an N-way directional coupler that may be used in the device in  FIG. 13 . This coupler is implemented by using a coupled transmission line  1401  that includes N transmission lines  1405  that are in parallel with each other. The length and width of each transmission line  1405  and the spacing between two adjacent transmission lines  1405  can be selected and optimized to satisfy the magnitude and phase requirements for eliminating unwanted coupling signals from the adjacent antenna elements (Ant 1  . . . AntN)  1301  as shown  FIG. 13 .  
       FIG. 15  illustrates an exemplary implementation of an N-way directional coupler utilizing metamaterial technology. The N-way metamaterial directional coupler can be constructed by using a coupled metamaterial transmission line  1520  which includes N CRLH metamaterial transmission lines (CRLH-TLs)  1505 - 1 ,  1505 - 2 ,  1505 - 3  that are in parallel with each other. N−1 additional coupling capacitors ( 1535 - 1 ,  1535 - 2 ,  1535 - 3 ), or collectively referred as C m s, are provided and each is connected between two adjacent CRLH-TLs to enhance the coupling. Each CRLH-TL ( 1505 - 1 ,  1505 - 2 ,  1505 - 3 ) in this example includes a series capacitor (C L1 , C L2 , C LN ), a shunt inductor (L L1 , L L2 , L LN ), and a section of a transmission line (TL 1 , TL 2 , TLN), respectively. The transmission lines ( 1501 - 1 ,  1501 - 2 ,  1501 - 3 ), TL 1  . . . TLN, from each CRLH-TLs, form a coupled transmission line which also contributes to the coupling between adjacent ports. For each metamaterial transmission line (CRLH-TL) ( 1505 - 1 ,  1505 - 2 ,  1505 - 3 ), the series capacitor C LN , ( 1530 - 1 ,  1530 - 2 ,  1530 - 3 ) and shunt inductor L LN , ( 1525 - 1 ,  1525 - 2 ,  1525 - 3 ), can have values that are different from each other. Factors related to the transmission line (TL) section ( 1501 - 1 ,  1501 - 2 ,  1501 - 3 ) that can be tuned to optimize the coupled transmission line, the input impedance, the coupling level between the adjacent ports, and the frequency where maximum coupling occurs may include, but are not limited to, width ( 1510 - 1 ,  1510 - 2 ,  1510 - 3 ), length  1530 , and spacing ( 1515 ) between adjacent transmission lines ( 1501 - 1 ,  1501 - 2 ,  1501 - 3 ), C m  ( 1535 - 1 ,  1535 - 2 ,  1535 - 3 ), C L  ( 1530 - 1 ,  1530 - 2 ,  1530 - 3 ), and L L  ( 1525 - 1 ,  1525 - 2 ,  1525 - 3 ). This can provide more free parameters in comparison to the conventional method to control the frequency response of the N-way directional coupler.  
      In the following sections, the two- and three-antennas systems demonstrate that the antenna performance, including isolation between antennas and radiation efficiencies, can be improved by incorporating a directional coupler. Such antenna performance improvements may contribute to boosting the communication system performances which may include, but are not limited to, channel capacity, coverage range, and bit error rate.  
      II. Exemplary Multi-Antenna Systems: Three-Element Antenna Array Coupled to Three-Way Directional Coupler  
       FIG. 16  illustrates an exemplary configuration of the three-antenna system  1600  which includes the three-element metamaterial antenna array  1601  and a three-way directional coupler  1620 , which is a subset of the generic multi-antenna system shown in  FIG. 13 . The three-way directional coupler  1620  can include three inputs  1615 , which are denoted as P 1 , P 2 , and P 3 . Three outputs  1610  of the directional coupler, P 4 , P 5 , and P 6 , can be connected to three antenna inputs  1605  of P 1 ′, P 2 ′ and P 3 ′, respectively. Of the Type I and Type II metamaterial antennas described in the example in  FIG. 17A  in this document, the Type I metamaterial antenna can be used for Ant 1  and Ant 3  while the Type II metamaterial antenna can be used for Ant 2  so that two adjacent antennas are made of different metamaterial types. The structure can be designed to make the coupling between Ant 1  and Ant 3  relatively small, and the coupling between Ant 1  and Ant 2  and that between Ant 3  and Ant 2  relatively large.  
      Details of various coupling between the inputs of the three-way directional coupler are described next. The input signal from P 1  can be coupled to P 2  through two paths. The first path starts at P 1  and proceeds to P 4  via the transmission of the directional coupler  1620 . Next, the signal from the output P 4  is transmitted to the antenna input P 1 ′ of Ant 1 . The signal radiated from Ant 1  can be coupled to Ant 2  which is also coupled to the antenna input P 2 ′. The signal at P 2 ′ is transmitted to P 5  and then proceeds through the transmission of the directional coupler  1620  from P 5  to P 2 . The second path starts at P 1  and ends at P 2  via the coupling of the directional coupler  1620 . When the coupled signals from the two paths merge at P 2  with the same magnitude and 180° phase difference, the two coupled signals may cancel each other out. This condition generally indicates that the isolation between P 1  and P 2  can be maximized. The input signal from P 3  can be coupled to P 2  through two paths. The first path starts at P 3  and proceeds to P 6  via the transmission of the directional coupler  1620 . Next, the signal from the output P 6  is transmitted to the antenna input P 3 ′ of Ant 3 . The signal radiated from Ant 3  is coupled to Ant 2  which is also coupled to the antenna input P 2 ′. The signal at P 2 ′ is transmitted to P 5  and then proceeds through the transmission of the directional coupler  1620  from P 5  to P 2 . The second path starts at P 3  and ends at P 2  via the coupling of the directional coupler. When the coupled signals from the two paths merge at P 2  with the same magnitude and 180° phase difference, the two coupled signals may cancel each other out. This condition generally indicates that the isolation between P 3  and P 2  can be maximized. In addition, the input signal from P 1  can be coupled to P 3  through two paths. The first path starts at P 1  and proceeds to P 4  via the transmission of the directional coupler  1620 , and the signal from the output P 4  is transmitted to the antenna input P 1 ′ of Ant 1 . The signal radiated from Ant 1  is coupled to Ant 3  which is also coupled to the antenna input P 3 ′. The signal at P 3 ′ is transmitted to P 6  and then proceeds through the transmission of the directional coupler  1620  from P 6  to P 3 . The second path starts at P 1  and ends at P 3  via the coupling of the directional coupler  1620 . Therefore, to preserve the high isolation between Ant 1  and Ant 3 , the coupling between P 1  and P 3  through the three-way directional coupler  1620  should be minimized.  
      II.A. Three-Element Metamaterial Antenna Array  
      Multiple antennas can be integrated in a single wireless device by using metamaterial technology.  FIGS. 17A-17B  and  FIG. 18  depict an exemplary implementation of a three-element metamaterial antenna array.  FIG. 17A  represents the top metal layer,  FIG. 17B  shows the bottom metal layer. The metamaterial antenna array  1700  shown in  FIG. 17A  includes three antennas, antennas  1701 - 1  and  1701 - 2  being made of the Type I metamaterial structure, and the other  1703  being made of the Type II metamaterial structure. Each antenna is coupled to an antenna CPW feed  1712  to send or receive a signal. The width  1740 , length  1745 , and gap  1750  of the antenna CPW feed  1712  are 1.1 mm, 17.65 mm, and 0.35 mm, respectively. The feed  1712  may also be implemented in a non-CPW design.  
       FIG. 18  shows a 3-Dimensional perspective view of a three-element metamaterial antenna array having the top layer  1804 , bottom layer  1812  and the substrate  1820 . All three antennas  1701 - 1 ,  1701 - 2  and  1701 - 3  in  FIGS. 17A and 17B  can be placed at one periphery on top of the substrate as shown in  FIG. 18 . In  FIG. 18 , the dimension, thickness, and dielectric constant of the substrate  1820  are 30 mm×55.56 mm, 0.787 mm, and 4.4, respectively. The two Type I antennas ( 1802 - 1  and  1802 - 2 ) can be placed at two sides on top of the substrate  1820  and may be symmetric with respect to the Type II antenna ( 1803 ). The Type II antenna  1803  may be located at the middle with respect to the substrate  1820 . Although Type I ( 1802 - 1  and  1802 - 2 ) and Type II ( 1803 ) antennas have different shapes. All three antennas  1801 - 1 ,  1801 - 2  and  1801 - 3  can be designed to operate at the same frequency band. Each antenna can be fed by a 50Ω conductor backed coplanar waveguide (CPW) feed  1805 . Also depicted in  FIG. 18  are a CPW ground on the top layer  1804 , launch pads  1810  on the top layer  1804 , cell patches  1815  on the top layer  1804 , a CPW ground  1825  located on the bottom layer  1812 , vias  1830  located on the substrate  1820 , via pads  1845  located on bottom layer  1812 , and via lines  1840  also located on the bottom layer  1812 .  
      Exemplary geometries and dimensions are described below with reference to  FIGS. 17A-17B  and  FIG. 18 . The two Type I antennas ( 1701 - 1  and  1701 - 2 ) are constructed identically, and have identical dimensions. Referring again to  FIG. 17A , the Type I metamaterial antenna  1701 - 1  can include a cell patch  1705 , a launch pad  1715 , a via  1710 , a via pad (shown in  FIG. 17B ) and a via line (shown in  FIG. 17B ). The cell patch  1705  of the Type I metamaterial antenna can be horizontally divided into an upper rectangular patch and a lower rectangular patch of different dimensions. In the illustrated example, the lower rectangular patch is smaller than the upper rectangular patch. Exemplary dimensions of the two rectangular patches are 4.9 mm×5.8 mm for the upper patch and 2.45 mm×1.5 mm for the lower patch. The cell patch  1705  can be coupled to the launch pad  1715  through a coupling gap  1738  which is about 0.2 mm×5.8 mm. The launch pad  1715  can include two vertically connected rectangular portions: an upper portion and a lower portion. For the Type I metamaterial antenna  1701 - 1 , the upper portion of the launch pad  1715  can be coupled to the cell patch  1705 , and the lower portion of the launch pad  1715  can be connected to the antenna CPW feed  1712 . Exemplary dimensions of the upper and lower portions of the launch pad  1715  are 0.8 mm×5.8 mm and 0.4 mm×2.3 mm, respectively. The cell patch  1705  can be connected to via pad  1770  of  FIG. 17B  on the bottom layer of the substrate  1820  of  FIG. 18  by using a metallic via  1775 . Now, referring to  FIGS. 17A-17B , the via  1775  is located at 7.37 mm away from the top of the cell patch  1705  edge portion and 1.40 mm away from the side edge portion of the substrate. The radius of the via  1710  in  FIG. 17A  is about 0.127 mm. The via pad  1770  in  FIG. 17B  of the Type I metamaterial antenna  1760 - 1  is 0.8 mm×0.8 mm and may be connected to the CPW ground  1763  through the via line  1780 . For the Type I metamaterial antenna  1760 - 1 , the via line  1780  can include two rectangular strips forming an L-shape strip. One strip of the via line  1780  can be coupled to via pad  1770 . Exemplary sizes for the one strip of the via line  1780  are 0.3 mm in width and 3.8 mm in length. The other strip of the via line  1780  can be connected to the CPW ground  1763 . Measurements for the other strip of the via line  1780  can be 0.3 mm in width and 5.25 mm in length. Two cut corners ( 1796 - 1 ,  1796 - 2 ) of the CPW ground  1763  in close proximity to the Type I metamaterial antenna may be cut on both the top and bottom layers of the substrate as shown in  FIGS. 17A-17B . The dimension of the rectangular cut is 2.95 mm×1 mm.  
      The Type II metamaterial antenna  1703  in  FIG. 17A  has a different geometry from the Type I metamaterial antenna  1701  and can include a cell patch  1725 , a launch pad  1735 , a via  1730 , a via pad (shown in  FIG. 17B ) and a via line (shown in  FIG. 17B ). The cell patch  1725  of the Type II metamaterial antenna  1703 , which is generally rectangular in shape and is 4.7 mm×7.0 mm, can be coupled to the launch pad  1735  through a coupling gap  1726  which is 4.7 mm×0.16 mm. The launch pad  1735  may include two vertically connected rectangular portions: an upper portion and a lower portion. The upper portion of the launch pad  1735  can be coupled to the cell patch  1725  via a gap, and the lower portion of the launch pad  1735  can be connected to the 50Ω antenna CPW feed  1712 . Exemplary dimensions of the upper and lower portions of the launch pad  1735  are 4.7 mm×1.5 mm and 0.4 mm×3.2 mm, respectively. The cell patch  1725  of  FIG. 17A  can be connected to the via pad  1790  of  FIG. 17B  on the bottom layer of the substrate  1820  of  FIG. 18  by using a metallic via  1795 . Referring to  FIGS. 17A-17B , the via  1795  may be located at 3.76 mm away from the top of the cell patch  1725  edge and 2.35 mm away from the cell patch  1725  side edge. The radius of the via  1795  in  FIG. 17B  can measure 0.127 mm. The via pad  1790  can be coupled to the CPW ground  1763  through the via line  1785 . A typical dimension for the via pad  1790  of Type II metamaterial antenna  1765  can be 0.6 mm×0.6 mm. The via line  1785  can be formed by a rectangular shape strip that has a dimension of 0.2 mm×7.8 mm.  
       FIG. 19  illustrates the simulation results of the three-element metamaterial antenna array shown in  FIGS. 17A-17B  and  FIG. 18 . Notably, the bandwidth within which the return loss is better than −10 dB for the Type I metamaterial antennas can range from about 2.46 GHz to 2.6 GHz as indicated by the simulated values for |S 1 ′ 1 ′. The coupling between the two Type I metamaterial antennas can be less than −13 dB across the entire above mentioned bandwidth as indicated by the simulated values for |S 1 ′ 3 ′|. Also from  FIG. 19 , the return loss for the Type II metamaterial antenna may be better than −10 dB from about 2.48 GHz to 2.55 GHz (as indicated by the simulated values for |S 2 ′ 2 ′|. The coupling between the Type II metamaterial antenna and Type I metamaterial antennas can be between −8 dB to −6 dB in the range of about 2.43 GHz to 2.6 GHz as shown by the simulated values of |S 1 ′ 2 ′|.  
      II.B1 Three-Element Antenna Array with Three-way Directional Coupler Using Microwave Coupled Lines  
      In  FIGS. 17A and 17B , the three-element metamaterial antenna array can be symmetric with respect to the center of the substrate. Thus, the structure of the three-way directional coupler should also be symmetric. One way to construct the three-way directional coupler is the use of microwave coupled line coupler. A directional coupler can be a four port device built by utilizing a microwave coupler which can have two transmission lines that are parallel to each other. In another embodiment, additional transmission lines are included to form a six-port three-way directional coupler.  
       FIG. 20  illustrates a structure of the three-way directional coupler  2000  with six ports (P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ), formed on a substrate  2020  such as FR-4. Exemplary values for thickness and dielectric constant of the FR-4 substrate are 0.787 mm and 4.4, respectively. The three-way directional coupler  2000  includes a CPW coupled line  2001 , CPW ground electrodes  2005 - 1  and  2005 - 2  formed in the same top metallization layer in which the CPW coupled line  2001  is formed and the CPW ground electrode  2005 - 3  in the bottom metallization layer. The CPW coupled line  2001  can, for example, include three microstrip lines  2025  that are arranged in parallel to each other and separated by a gap  2035 . The width  2030 , w, of a single microstrip line  2010  may be 1.1 mm and the gap width  2035 , s, may be 0.1 mm as shown in  FIG. 20 . Under this configuration, to maximize the coupling at a frequency of 2.52 GHz, the length of the CPW coupled line  2001  can be set to 16.9 mm. The distance between the CPW coupled line and the top portion of the CPW ground is denoted by “g”  2040  in  FIG. 20  and measures 0.75 mm in width.  
       FIG. 21  and  FIG. 22  show the simulated results of the three-way directional coupler  2000  in  FIG. 20  and indicate all six ports of the three-way directional coupler  2000  are matched to 50Ω. The low insertion losses between P 1  and P 4  (|S 41 |), P 2  and P 5  (|S 52 |), and P 3  and P 6  (same as |S 41 |) are obtained. The maximum coupling of −9.3 dB between P 1  and P 2  (|S 21 |) and P 3  and P 2  (|S 32 |) or P 4  and P 5  (same as |S 21 |) and P 6  and P 5  (same as |S 32 |) occurs at around 2.5 GHz. The coupling between P 1  and P 3  (|S 31 |) and P 4  and P 6  (same as |S 31 |) is less than −20 dB from the range of about 1 GHz to 4 GHz. These results generally satisfy the requirements of a high coupling between (P 1  and P 2 ), (P 4  and P 5 ), (P 2  and P 3 ), and (P 5  and P 6 ) and a low coupling between (P 1  and P 3 ) and (P 4  and P 6 ).  
       FIGS. 23A, 23B , and  24  show a specific exemplary implementation of the three-antenna system illustrated in  FIG. 16  with a three-element metamaterial antenna array and a three-way directional coupler, which is a subset and en example of the multi-antenna system shown in  FIG. 13 . The dimensions of the Type I and Type II metamaterial antennas shown in  FIGS. 23A, 23B , and  24  may be implemented to be the same as the three-element metamaterial antenna array shown in  FIGS. 17A-17B  and  FIG. 18  with the exception of the antenna CPW feed lines.  FIG. 23A  represents a top layer,  FIG. 23B  represents a bottom layer, and  FIG. 24  represents a 3-Dimensional stacked view of the top layer  2403 , bottom layer  2432  and a substrate  2425  of the three-element metamaterial antenna array. The length of the antenna CPW feed  2320  shown in  FIG. 23A  can be optimized to satisfy the phase requirement as previously indicated.  
      With respect to the Type I metamaterial antenna  2302  shown on the left-hand side of  FIG. 23A , one end portion of an antenna CPW feed  2320 - 1  is connected to a CPW coupled line  2340  via a CPW adjoining line  2330 - 1 . The antenna CPW feed  2320 - 1  and the CPW adjoining line  2330 - 1  form an L-shape structure. The adjoining line  2330 - 1  can include two CPW bends: a first bend  2325 - 1  and a second bend  2325 - 2 . The first bend  2325 - 1  is connected to the antenna CPW feed  2320 - 1 , and the second bend  2325 - 2  which is connected to the CPW coupled line  2340 . The other end portion of the antenna CPW feed  2320 - 1  is connected to the launch pad  2315 - 1  of the left-hand side of the Type I metamaterial antenna  2302 . For example, the antenna CPW feed  2320 - 1  may 1.1 mm×18 mm, and the CPW adjoining line  2330 - 1  may be 6.9476 mm×1.1 mm. The two CPW bends ( 2325 - 1 ,  2325 - 2 ) can form a triangle, and the dimensions of the two sides that form the right angle can be 1.1 mm.  
      For the Type I metamaterial antenna  2304  shown on the right-hand side of  FIG. 23A , the antenna CPW feed  2320 - 3  and the CPW adjoining line  2330 - 2  structure form a mirrored L-shape structure that is identical in structure and dimensions to the L-shaped structure of the Type I metamaterial antenna  2302  formed on the left-hand side. The antenna CPW feed  2320 - 2  connected to the Type II metamaterial antenna  2303  may be 1.1 mm×19.1 mm in dimension. The structure of the CPW coupled line  2340  is identical to the three-way directional coupler  2000  shown in  FIG. 20  and the dimensions are the same as previously indicated.  
      Input ports, P 1 , P 2 , and P 3 , of the CPW feed lines CPW 1   2350 , CPW 2   2355 , and CPW 3   2360  are connected to the CPW coupled line  2340  in which CPW 1   2350 , CPW 2   2355 , and CPW 3   2360  form a CPW feed  2345  as shown in  FIG. 23A . CPW 1   2350  and CPW 3   2360  each have a dimension of 3 mm×1.1 mm, and each are connected to one end portion of the CPW coupled line  2340  via CPW bends  2337 - 1  and  2337 - 2  respectively. The CPW bends ( 2337 - 1 ,  2337 - 2 ) may be identical to the first  2325 - 1  and second  2325 - 2  CPW bends mentioned above. The CPW 2   2355  is connected to the middle portion of the CPW coupled line  2340  and may have a dimension of 1.1 mm×3 mm. Other components shown in  FIGS. 23A-23B  have been covered in the previous sections which include cell patch  2301 , via ( 2310 - 1 ,  2310 - 2 ,  2310 - 3 ), launch pad ( 2315 - 1 ,  2315 - 2 ,  2315 - 3 ), via line  2370  and CPW ground  2335 .  
       FIG. 24  depicts a 3-Dimensional stacked view and alignment of the top layer  2403  and the bottom layer  2432  which are also depicted in detail in  FIGS. 23A-23B , respectively. Specifically, the components shown in  FIG. 24  show a 3-D rendering of the same components depicted in  FIGS. 23A-23B  which include cell patch  2401 , launch pad  2405 , CPW coupled line  2410 , CPW feed  2415 , CPW ground ( 2420 ,  2430 ), substrate  2425 , via  2427 , via pad  2437 , and via line  2433 .  
       FIG. 25  shows simulation results of the three-antenna system above by using Ansoft HFSS. Notably, the isolation between P 1  and P 3  is preserved to be less than −10 dB and the isolations between (P 1  and P 2 ) and (P 3  and P 2 ) are improved in comparison to the results shown in  FIG. 19 . The measured radiation efficiencies of three antenna system shown in  FIG. 24  are illustrated in  FIG. 26 . Thus, by improving the isolation of the Type II metamaterial antenna, greater radiation efficiency can be achieved as shown in  FIG. 26 .  
      II.B2 Three-Element Antenna Array with Three-Way Directional Coupler Using MTM Transmission Lines  
      An N-way directional coupler, e.g., a three-way directional coupler can be implemented based on the metamaterial technology to achieve a reduced circuit size with minimal adverse impact to circuit performance.  FIG. 27  illustrates an exemplary structure of a three-way MTM coupler  2700  which may be built on a 0.787 mm FR-4 substrate with a dielectric constant of 4.4. This three-way MTM coupler  2700  includes three CRLH metamaterial transmission lines (CRLH-TL 1   2701 , CRLH-TL 2   2702 - 1 , CRLH-TL 3   2702 - 2 ) that are parallel to each other. To enhance the coupling, a coupling capacitor ( 2730 - 1 ,  2730 - 2 ), C m , can be connected in between adjacent metamaterial transmission lines  2701 ,  2702 - 1  and  2702 - 2 . The metamaterial transmission line  2701  can be configured in a first configuration, and the other two metamaterial transmission lines,  2702 - 1  and  2702 - 2 , can be configured a second, different configuration. The configuration differences between CRLH-TL 1   2701  and CRLH-TL 2  ( 2702 - 1 ,  2702 - 2 ) can be used as parameters to optimize the three-way MTM coupler for impedance matching and phase adjustment purposes.  
      In example in  FIG. 27 , the CRLH-TL 1   2701  may include a section of a microstrip line  2716  (MCL 1 ), a series capacitor  2726  (C L1 ) and a shunt inductor  2722  (L L1 ). The CRLH-TL 2  may include a section of a microstrip line  2715 - 1  or  2715 - 2  (MCL 2 ), a series capacitor  2725 - 1  or  2725 - 2  (C L2 ), and a shunt inductor  2720 - 1  or  2720 - 2  (L L2 ). In one implementation, each of the microstrip lines  2716 ,  2715 - 1  and  2715 - 2  can be the right-handed portion of the respective CRLH-TL  2701 ,  2702 - 1  or  2702 - 2 , and the lumped elements generally represent the left-handed portion of the respective CRLH-TL  2701 ,  2702 - 1  or  2702 - 2 . For example, the width w 1   2712  and length L 1   2718  of the microstrip line section  2716 , MCL 1 , may be 0.5 mm and 4 mm, respectively. The series capacitor  2726 , C L1 , and shunt inductor  2722 , L L1 , may be 8 pF and 2.3 nH, respectively. The width w 2  ( 2710 - 1 ,  2710 - 2 ) and length L 2  ( 2705 - 1 ,  2705 - 2 ) of the microstrip line section ( 2715 - 1 ,  2715 - 2 ), MCL 2 , may be 1.9 mm and 4 mm, respectively. The series capacitor ( 2725 - 1 ,  2725 - 2 ), C L2 , and shunt inductor ( 2720 - 1 ,  2720 - 2 ), L L2 , may be 15 pF and 2.9 nH, respectively.  
      To construct the three-way MTM coupler, the three metamaterial transmission lines ( 2701 ,  2702 - 1 ,  2702 - 2 ) can be arranged in parallel and in the order of CRLH-TL 2   2702 - 1 , CRLH-TL 1   2701  and CRLH-TL 2   2702 - 2 . The three microstrip line sections, which can include one MCL 1   2716  and two MCL 2 &#39;s ( 2715 - 1 ,  2715 - 2 ), form a three-way microstrip coupled line  2703  which may contribute to the coupling between adjacent metamaterial transmission lines. The spacing, s ( 2719 - 1 ,  2719 - 2 ), between each microstrip line section, MCL 1   2716  and MCL 2  ( 2715 - 1 ,  2715 - 2 ), may be 0.1 mm, and the capacitance of the coupling capacitor, C m  ( 2730 - 1 ,  2730 - 2 ) may be 1 pF. Ports P 1 , P 2 , P 3 , P 4 , P 5 , and P 6  are I/O ports and are capable of either receiving or transmitting a signal of the three-way MTM coupler  2700 .  
       FIG. 28  shows the simulated S-parameters for the input signal at P 1  of  FIG. 27 . Due to the symmetric configuration of the MTM coupler shown in  FIG. 27 , the same results can be obtained for P 3 , P 4 , and P 6  as well. The results suggest a good impedance matching in the range of about 1.85 GHz to 4 GHz with a return loss of better than −10 dB. A high coupling may occur between P 1  and P 2  (P 3  and P 2 , P 4  and P 5 , P 6  and P 5 ) in a frequency range of about 2.4 GHz to 2.7 GHz. As can be expected, the coupling between P 1  and P 3  (P 4  and P 6 ) is low.  
       FIG. 29  illustrates the simulated S-parameters for the input signal at P 2 . The same results can be obtained for P 5  as well. The results indicate an impedance matching with a return loss of better than −10 dB in the range of about 2 GHz to 4 GHz. A high coupling occurs between (P 2  and P 1 ) and (P 2  and P 3 ) and between (P 5  and P 4 ) and (P 5  and P 6 ) in a frequency range of about 2.4 GHz to 2.7 GHz.  
      The three-antenna system can be constructed by combining the three-element metamaterial antenna array shown in  FIGS. 17A-17B  and the three-way MTM coupler  2700  shown in  FIG. 27 . The three-way MTM coupler  2700  include output ports P 4 , P 5 , and P 6  (from  FIG. 27 ) and can connect to the three-element metamaterial antenna array input ports P 1 ′, P 2 ′ and P 3 ′ (from  FIG. 17A ), respectively. The dimensions and the lumped element values associated with the three-way MTM coupler  2700  can be further optimized to satisfy the magnitude and phase requirements for eliminating unwanted coupling signals from the adjacent antenna elements as discussed in the previous sections. In one optimized example where the magnitude and phase requirements are met, the width  2712  and length  2718  of the CRLH-TL 1  microstrip line (MCL 1 )  2716  section shown in  FIG. 27  are 0.8 mm and 5 mm, respectively. The series capacitor  2726 , C L1 , and a shunt inductor  2722 , L L1 , for CRLH-TL 1   2701  are 18 pF and 2.5 nH, respectively. The width ( 2710 - 1 ,  2710 - 2 ) and length ( 2705 - 1 ,  2705 - 2 ) of the microstrip line (MCL 2 ) ( 2715 - 1 ,  2715 - 2 ) section are 1.8 mm and 5 mm, respectively. The series capacitor ( 2725 - 1 ,  2725 - 2 ), C L2 , and a shunt inductor ( 2720 - 1 ,  2720 - 2 ), L L2 , for CRLH-TL 2  ( 2702 - 1 ,  2702 - 2 ) are 8 pF and 3 nH, respectively. In addition, the spacing, s ( 2719 - 1 ,  2719 - 2 ), between adjacent microstrip line sections, MCL 1   2716  and MCL 2  ( 2715 - 1 ,  2715 - 2 ), is 0.1 mm, and the capacitance of the coupling capacitor ( 2730 - 1 ,  2730 - 2 ), C m , is 1.2 pF.  
       FIG. 30  illustrates the simulated results of the three-antenna system using three-way MTM coupler  2700  in  FIG. 27 . The impedance matching is maintained as in the case of the three-element metamaterial antenna array shown in  FIGS. 17A-17B ,  18 ,  19 . The high isolation between P 1  and P 3  is also retained as predicted. A comparison between  FIG. 30  and  FIG. 19  indicates that an improved isolation between (P 1  and P 2 ) or (P 2  and P 3 ) can be achieved. This isolation improvement can lead to higher radiation efficiency as discussed in the previous section.  
      III. Single-Band Multi-Antenna System: Two-Element Antenna Array with 2-Way Directional Coupler  
       FIG. 31A  and  FIG. 31B  illustrates an exemplary configuration of a two-antenna system  3100 -A and  3100 -B which includes a two-element metamaterial antenna array (including Ant 1   3101  and Ant 2   3105 ) and a two-way directional coupler  3130 , which is a subset of the multi-antenna system shown in  FIG. 13 . The two-way directional coupler  3130  can include two inputs  3135  and  3140 , which are denoted as P 1  and P 2 , respectively. Two outputs, P 3   3120  and P 4   3125 , of the directional coupler, can be connected to two antenna inputs P 1 ′  3110 , P 2 ′  3115 , respectively.  
      A detailed description of coupling between the inputs of the directional coupler is presented next. The input signal from P 1   3135  can be coupled to P 2   3140  through two paths. The first path starts at P 1   3135  and proceeds to P 3   3120  via the transmission of the directional coupler  3130 . Next, the signal from the output P 3   3120  is transmitted to the antenna input P 1 ′  3110  of Ant 1   3101 . The signal radiated from Ant 1   3101  can be coupled to Ant 2   3105  which is also coupled to the antenna input P 2 ′  3115 . The signal at P 2 ′  3115  is transmitted to P 4   3125  and then proceeds through the transmission of the directional coupler  3130  from P 4   3125  to P 2   3140 . The second path starts at P 1   3135  and ends at P 2   3140  via the coupling of the directional coupler  3130 . When the coupled signals from the two paths merge at P 2   3140  with the same magnitude and 180° phase difference, the two coupled signals may cancel each other out. This condition generally maximizes the isolation between P 1   3135  and P 2   3140 .  
      III.A1 Single-Band Two-Element Antenna Array with Two-way Directional Coupler Using Microwave Coupled Lines  
      Multiple views showing various layers and elements of the multi-antenna system are depicted in  FIGS. 32A-32D . For example,  FIG. 32A  shows the 3-dimensional view of stacked layers forming the multi-antenna system.  FIG. 32B  depicts the top layer of the multi-antenna system which comprises two-antenna elements.  FIG. 32C  depicts the bottom layer of the multi-antenna system, and  FIG. 32D  depicts a cross-sectional view of the multi-antenna system.  
      Referring again to  FIG. 31A , the multi-antenna system  3100  can include the two-element antenna array ( 3101 ,  3105 ) and the two-way directional coupler  3130  which can be implemented by using a metamaterial antenna array  3300 , as shown in  FIG. 33 , and a microwave directional coupler  3400 , as shown in  FIG. 34 , respectively. A detailed description of each element is presented in Table 2.  
      In one implementation of the device in  FIG. 33 , the multi-antenna system  3100  in  FIG. 31A  can be designed on a 1-mm FR4 substrate with a dielectric constant of 4.4. The Ant 1   3303 - 1  may be fed by a 50Ω microstrip feed line  3310 - 1  which may have a dimension of 1.4 mm×20 mm. One side of the 50Ω microstrip feed line  3310 - 1  may be directly connected to a launch pad  3301 - 1  of the Ant 1   3303 - 1  while the other side of the 50Ω microstrip feed line  3310 - 1  may be connected to the input port P 1 ′  3315 - 1 . In this example, the launch pad  3301 - 1  may include two rectangular shape lines. The dimension of the first rectangular shape line, which is connected to the 50Ω microstrip feed line  3110 - 1 , may have a dimension of 0.4 mm×3.2 mm while the other line is capacitively coupled to the cell patch  3340 - 1  through a coupling gap  3325 - 1  (e.g., 0.16 mm) and may have a dimension of 4.7 mm×1.5 mm. The cell patch  3340 - 1  is shorted to the microstrip ground  3320  through a via  3330 - 1 , a via pad  3335 - 1  and a ground line  3305 - 1 . The cell patch  3340 - 1 , in this example, may have a dimension of 4.7 mm×7 mm. The via  3330 - 1  is connected to the cell patch  3340 - 1  on one side of the substrate and to the via pad  3335 - 1  on the opposing side of the substrate. The via  3330 - 1  may have a radius of 0.15 mm and may be located at 2.96 mm from the top open end portion of the cell patch  3340 - 1  to the center of the via  3330 - 1 . The via pad  3335 - 1  may have a dimension of 0.6 mm×0.6 mm and is connected to the microstrip ground  3320  through a ground line  3305 - 1 . The dimension of the ground line  3305 - 1  may be 0.2 mm×8.6 mm. For the metamaterial antenna Ant 2   3303 - 2 , dimensions may be the same as the Ant 1   3303 - 1 . The spacing between the inside edge portion of the Ant 1   3303 - 1  and the inside edge portion of the Ant 2   3303 - 2  may be about 13 mm. Elements for Ant 2   3303 - 2  include a cell patch  3340 - 2 , via  3330 - 2 , via pad  3335 - 2 , coupling gap  3325 - 2 , 50Ω microstrip feed line  3310 - 2 , ground line  3305 - 2 , port P 2 ′  3315 - 2 , and launch pad  3301 - 2 .  
      Referring to  FIG. 34 , the microwave directional coupler  3400  has four input/output ports (P 1   3405 - 1 , P 2   3405 - 2 , P 3   3405 - 3 , and P 4   3405 - 4 ) where ports P 1   3405 - 1  and P 2   3405 - 2  can be used for the RF inputs while ports P 3   3405 - 3  and P 4   3405 - 4  are the outputs of the microwave directional coupler  3400 , which can be connected to the metamaterial antenna array  3300  of  FIG. 33 . The dimension of each 50Ω microstrip feed line  3401  at the input end may have a dimension of 1.48 mm×5 mm, while the dimension of each microstrip feed line  3435  at the output end may be a 50Ω element and may have a dimension of 1.4 mm×2.15 mm. The coupling portion of the microwave directional coupler  3400  is realized by a microstrip coupled line  3420  where the length, width and coupling gap  3415  of the microstrip coupled line  3420  may be 14 mm, 0.4 mm and 0.1 mm, respectively. Four ends of microstrip coupled line  3420  are connected to four 50Ω microstrip feed line ( 3401 ,  3435 ) through four microstrip tapered lines ( 3410 - 1 ,  3410 - 2 ,  3410 - 3 ,  3410 - 4 ) and microstrip bends ( 3425 - 1 ,  3425 - 2 ) for the impedance matching purpose. The length, L 1   3436 , of the microstrip tapered line  3410 - 2  that is connected to the P 3   3405 - 3 , may be 5.35 mm. The widths, w 21   3437 - 1  and w 22   3437 - 2 , of the microstrip tapered line  3410 - 2  may be 1.4 mm and 0.4 mm, respectively. The corresponding length and widths of the microstrip tapered line  3410 - 3  have the same dimensions as the microstrip tapered line  3410 - 2 . The length, L 2   3438 , of the microstrip tapered line  3410 - 1  that is connected to the P 1   3405 - 1 , may be 8.9 mm. The widths, w 11   3439 - 1  and w 12   3439 - 2 , of the microstrip tapered line  3410 - 1  may be 1.48 mm and 0.4 mm, respectively. The corresponding length and widths of the microstrip tapered line  3410 - 4  can have the same dimensions as the microstrip tapered line  3410 - 1 .  
      The multi-antenna system shown in  FIGS. 32A-32D  is simulated by using Ansoft HFSS. Designs are fabricated and tested using a network analyzer.  FIG. 35  illustrates the return losses of the two metamaterial antenna elements ( 3303 - 1  and  3303 - 2 ) and coupling level between the two metamaterial antenna elements ( 3303 - 1 ,  3303 - 2 ) in  FIG. 33 .  FIG. 36  illustrates the return losses of the multi-antenna system shown in  FIGS. 32A-32D  and the coupling level at inputs (P 1   3405 - 1  and P 2   3405 - 2 ), shown in  FIG. 34  when P 3   3405 - 2  and P 4   3405 - 4  are connected to metamaterial antenna elements ( 3303 - 1 ,  3303 - 2 ) in  FIG. 33 . Based on these results, the isolation between the two MTM antenna elements ( 3303 - 1 ,  3303 - 2 ) of  FIG. 33  can be improved while maintaining a low return loss and a sufficient bandwidth.  
       FIGS. 37A-37C  illustrate radiation patterns of the multi-antenna system of  FIGS. 32A-32D . Notably, radiation beam patterns shown in  FIGS. 37A-37C  point in opposite directions allowing the two signals to propagate in different paths. Such results generally indicate successful pattern diversity and low far-field envelope correlation in the multi-antenna system of  FIGS. 32A-32D .  
       FIG. 38A  shows a fabricated multi-antenna system of  FIGS. 32A-32D  while  FIG. 38B  depicts the measured return losses and isolation.  FIG. 39  illustrates a comparison of the measured radiation efficiencies for the multi-antenna system with (shown in  FIGS. 32A-32D ) and without (shown in  FIG. 33 ) the microwave directional coupler  3400  as shown in  FIG. 34 . The efficiency with the microwave directional coupler  3400  is increased by around 10% at about 2.4 GHz.  
               TABLE 2                          Multi-Antenna, Directional Coupler System: Two-Element       Antenna Array, Two-way Directional Coupler using Microwave       Coupled Lines (single band)                         Parameter   Description   Location               Multi-   Multi-antenna system includes a           Antenna   Metamaterial Antenna Array and a       System   Microwave Directional Coupler.       Metamaterial   Antenna array comprises two MTM       Antenna   Antenna Elements.       Array       MTM Antenna   Each antenna element comprises an MTM       Element   Cell coupled to the 50 Ω microstrip           line via a Launch Pad. Launch Pad is           located on top of the substrate.       Launch Pad   Two rectangular shape that connects   Top Layer           Cell Patch to the 50 Ω microstrip feed           line. There is a coupling gap between           the Launch Pad and the Cell Patch.                             MTM Cell   Cell   Rectangular shape   Top Layer           Patch           Via   Cylindrical shape and connects   Top Layer               the Cell Patch with the Via   to Bottom               Pad.   Layer           Via   Small square pad that connects   Bottom           Pad   the bottom part of the Via to   Layer               the GND Line.           GND   Connects the Via Pad to the   Bottom           Line   main GND   Layer                         Microwave   Directional coupler includes a           Directional   Microstrip Coupled Line, four Tapered       Coupler   Lines, and Four Microstrip Bend       Microstrip   Two parallel microstrip line with a   Top Layer       Coupled Line   coupling gap in between.       Tapered Line   Microstrip line with different line   Top Layer           width at both ends.       Microstrip   Triangular shape of microstrip   Top Layer       Bend   junction to connect two perpendicular           microstrip lines.                    
 III.A2 Single-Band Two-Element Antenna Array with Two-way Directional Coupler using MTM Transmission Line 
 
      In  FIG. 31A , the size of the multi-antenna system  3100  is dependent on the metamaterial antenna array ( 3101 ,  3105 ) and the microwave directional coupler  3130 . Therefore, the overall size of the multi-antenna system in  FIGS. 32A-32D  can be reduced by shrinking the coupler size. As shown in  FIGS. 40A-40D , a smaller multi-antenna system can be achieved where the microwave directional coupler  3400  of  FIG. 34  is replaced by an MTM coupler  4100  of  FIG. 41A , and the two MTM antenna array remains the same as in the previous implementation shown in  FIG. 33 .  FIG. 41B  and  FIG. 41C  show specific portions of the coupled transmission line and a pair of metamaterial transmission lines, respectively, in the same MTM coupler  4100  of  FIG. 41A . Each antenna element is presented in detail in Table 3.  
      A detailed view of the MTM coupler  4100  is presented in  FIG. 41A . The MTM coupler  4100  of  FIG. 41A  has four ports (P 1   4145 - 1 , P 2   4145 - 2 , P 3   4145 - 3 , P 4   4145 - 4 ) that can be used as input and output to the coupler. In this example, ports P 1   4145 - 1  and P 2   4145 - 2  can be used for the RF inputs while ports P 3   4145 - 3  and P 4   4145 - 4  can be used for the outputs of the MTM coupler, which can be connected to the two metamaterial antenna input ports P 1 ′  3315 - 1  and P 2 ′  3315 - 2  as shown in  FIG. 33 . The dimension of each 50Ω microstrip feed line  4101 - 1  for the two coupler inputs is 1.48 mm×5 mm, and the dimension of each 50Ω microstrip feed line  4101 - 2  for the two coupler outputs is 1.4 mm×3.15 mm.  
      To replace the microwave directional coupler  3400  of  FIG. 34  with MTM coupler  4100  of  FIG. 41A , the microstrip coupled line  3420  shown in  FIG. 34  can be replaced by using an MTM coupled line  4115  as shown in  FIG. 41B . The MTM coupled line  4115  shown in  FIG. 41B  can include two parallel MTM transmission lines ( 4116 - 1 ,  4116 - 2 ) as shown in  FIG. 41C . The MTM transmission line  4116 - 2  of  FIG. 41C  can include two microstrip lines sections ( 4115 - 2   a  and  4115 - 2   b ), capacitor pads  4127 , three series capacitors ( 4130 ,  4140 ) and two shorted stubs  4155  as shown in  FIG. 41A . The other MTM transmission line  4116 - 1  may have identical components as the MTM transmission line  4116 - 2 . The microstrip line sections ( 4115 - 1   a  and  4115 - 1   b ,  4115 - 2   a  and  4115 - 2   b ), in this implementation, can have the same dimensions where each of the line sections measures about 0.4 mm×2 mm.  
      The MTM coupler  4100  of  FIG. 41A  may include a coupling portion that is realized by an MTM coupled line  4115  of  FIG. 41B  where the two MTM transmission lines  4116 - 1  and  4116 - 2  of  FIG. 41C , can be placed in parallel with each other. In  FIG. 41A , a coupling capacitor Cm  4150  may be used to connect the two MTM transmission lines  4116 - 1  and  4116 - 2  of  FIG. 41C . The total length of the MTM coupled line  4115  shown in  FIG. 41B  is about 6.4 mm while the gap between the two MTM transmission lines  4116 - 1  and  4116 - 2  shown in  FIG. 41C  is about 1 mm. The coupling capacitor  4150  of 0.5 pF can be used in this implementation to enhance the coupling between the MTM transmission lines ( 4116 - 1  and  4116 - 2 ) shown in  FIG. 41C .  
      Referring again to  FIG. 41A , two microstrip line sections  4115 - 2   a  and  4115 - 2   b  can be connected by three series capacitors in the sequence of 2C L    4130 , C L    4140 , and 2C L    4130 . Two capacitor pads  4127  located between the two microstrip line sections  4115 - 2   a  and  4115 - 2   b  can be used as metal bases to mount the series capacitors ( 4130 ,  4140 ) on. In one implementation, C L    4140  is realized by using the chip capacitor with value of 0.85 pF and 2C L  is realized by using the chip capacitor with value of 1.7 pF. The spacing between the microstrip line section ( 4115 - 2   a  and  4115 - 2   b ) and the capacitor pad  4127  is about 0.4 mm. The spacing between the two capacitor pads  4127  is also about 0.4 mm. Each capacitor pad  4127  has a dimension of about 0.6 mm×0.8 mm. One side of the shorted stub  4155  can be attached at the center of the capacitor pad  4127  and the other side may be connected to the via pad  4120 . The via pad  4120  can be connected to the microstrip GND  4160  through the via  4125 . The shorted stub  4155  has a dimension of about 0.1 mm×3 mm. The via pad  4120  has a dimension of about 0.6 mm×0.6 mm. The via  4125  can be centered at the via pad  4120  having a radius of about 0.15 mm and height of about 1 mm. The four microstrip line sections ( 4115 - 1   a ,  4115 - 1   b ,  4115 - 2   a ,  4115 - 2   b ) may be connected to the four 50Ω microstrip feed lines ( 4101 - 1 ,  4101 - 2 ) through four microstrip tapered lines ( 4105 - 1   a ,  4105 - 1   b ,  4105 - 2   a ,  4105 - 2   b ) and four microstrip bends ( 4110 - 1   a ,  4110 - 1   b ,  4110 - 2   a ,  4110 - 2   b ) for impedance matching purpose. In  FIG. 41A , the length of microstrip tapered line ( 4105 - 1   a ,  4105 - 1   b ) that is connected to the 50Ω microstrip feed line  4101 - 1  measures about 8.35 mm while the widths of each microstrip tapered line ( 4105 - 1   a ,  4105 - 1   b ) measure about 1.48 mm at one end and about 0.4 mm at the other end. The length of each microstrip tapered line ( 4105 - 2   a ,  4105 - 2   b ) that is connected to the 50Ω microstrip feed line  4101 - 2  measures about 4.9 mm while the widths for each microstrip tapered line ( 4105 - 2   a ,  4105 - 2   b ) measure about 1.4 mm at one end portion and about 0.4 mm at the other end portion.  
      The multi-antenna system shown in  FIGS. 40A-40D  is simulated by using Ansoft HFSS while designs can be fabricated and tested using a network analyzer.  FIG. 42  shows the return losses and coupling level between two inputs of the multi-antenna system shown in  FIGS. 40A-40D  in which an improvement of the isolation between the two inputs is obtained as compared to the result shown in  FIG. 35 .  
       FIG. 43A-43C  illustrates radiation patterns of the multi-antenna system using the MTM coupler shown in  FIGS. 40A-40D  in which two opposite beam directions with respect to two inputs occur. Such results generally indicate successful pattern diversity and low far-field envelope correlation.  
       FIGS. 44A-44B  shows a fabricated multi-antenna system shown in  FIGS. 40A-40D  while  FIG. 44C  illustrates the measured return losses and isolation between two inputs of multi-antenna system shown in  FIGS. 40A-40D .  
       FIG. 45  shows a comparison of the measured radiation efficiencies for the multi-antenna system presented in this section with and without the MTM coupler  4100  shown in  FIG. 41A . In this implementation, the efficiency with MTM coupler is raised by about 15% at about 2.5 GHz.  
               TABLE 3                          Multi-Antenna, Directional Coupler System: Two-Element       Antenna Array, Two-way Directional Coupler using MTM       Transmission Line (single band)                         Parameter   Description   Location               Multi-   Multi-antenna system includes an MTM           Antenna   Antenna Array and an MTM Coupler.       System       MTM Antenna   Antenna array includes two MTM Antenna       Array   Elements.       MTM Antenna   Each antenna element includes an MTM Cell       Element   coupled to the 50 Ω microstrip line via           a Launch Pad. Launch Pad is located on           top of the substrate.       Launch Pad   Two rectangular shape that connects Cell   Top           Patch to the 50 Ω microstrip feed line.   Layer           There is a coupling gap between the           Launch Pad and the Cell Patch.                             MTM Cell   Cell   Rectangular shape   Top           Patch       Layer           Via   Cylindrical shape and   Top               connects the Cell Patch with   Layer to               the Via Pad.   Bottom                   Layer           Via Pad   Small square pad that   Bottom               connects the bottom part of   Layer               the Via to the Ground Line.           Ground   Connects the Via Pad to the   Bottom           Line   microstrip ground.   Layer                         MTM Coupler   Two MTM Transmission Lines parallel to               each other with Coupling Capacitor           connecting the two lines. Each MTM           Transmission Line includes two Microstrip           Line sections, Series Capacitors,           Capacitor Pad, Shorted Stub, via Pad, and           Via.                                 Microstrip   Rectangular shape line.   Top           Line       Layer           Series   Chip capacitor (2*CL) which   Top           Capacitor   connects one end of the   Layer               Microstrip Line and one end               of the Capacitor Pad. Chip               capacitor (CL) which connects               between two Capacitor Pads.           Coupling   Chip capacitor (Cm) which   Top           Capacitor   connects between two   Layer               Capacitor Pads in the               directional perpendicular to               the Microstrip Line.           Capacitor   Rectangular shape.   Top           Pad       Layer           Shorted   Rectangular shape line with   Top           Stub   one end connected to the   Layer               Capacitor Pad and the other               end connected to the Via Pad.           Via Pad   Square shape.   Top                   Layer           Via   Cylindrical shape. Connecting   Top               Via Pad to microstrip ground.   Layer                         Tapered   Microstrip line with different line width   Top       Line   at both ends.   Layer       Microstrip   Triangular shape of microstrip junction   Top       Bend   to connect two perpendicular microstrip   Layer           lines.                    
 III.A3 Single-Band Two-Element Antenna Array with MTM Transmission Line Feed and Two-way Directional Coupler Using MTM Transmission Line 
 
      To further reduce the overall size of the multi-antenna system of  FIGS. 40A-40D , shorter feed lines of the metamaterial antenna array can be utilized to reduce the size while still maintaining the same phase of the previous sections. In this implementation, the shorter feed lines of the metamaterial antenna array can be utilized to decouple the two input/output signals by either microwave directional coupler or the MTM coupler.  
       FIGS. 46A-46D  illustrates multiple views of layers and elements of the multi-antenna system presented in this section. In this implementation, the multi-antenna system may include a metamaterial antenna array with 13 mm spacing between the inner edges of two antenna elements and an MTM coupler. The multi-antenna system shown in  FIGS. 46A-46D  can be designed on a 1 mm FR4 substrate having a dielectric constant of 4.4.  
      A detailed view of a metamaterial antenna array  4700  and a MTM coupler  4800  are shown in  FIG. 47A  and  FIG. 48 , respectively.  FIG. 47B  represents the same metamaterial antenna array  4700  of  FIG. 47A  and outlines the specific portions of metamaterial transmission lines. Each element is described in Table 4.  
      In this example, a metamaterial transmission line ( 4736 - 1 ,  4736 - 2 ) shown in  FIG. 47B  is used instead of using microstrip feed line for the metamaterial antenna array  4700  shown in  FIG. 47A . The transmission line designed by metamaterial technology is known to have properties such that the propagation constant can be controlled to satisfy the phase requirement of the design while still maintaining a small physical size. Therefore, significant size reduction of the multi-antenna system can be achieved by using the metamaterial transmission line for the antenna feed.  
      Referring again to  FIG. 47A , one antenna element in the metamaterial antenna array includes an cell patch  4701 - 1  which is coupled to a launch pad ( 4710 - 1   a  and  4710 - 1   b ) through a coupling gap  4720 - 1 . The cell patch  4701 - 1  may have a dimension of about 4.7 mm×7 mm and the coupling gap  4720 - 1  may measure about 0.16 mm. The launch pad can include two rectangular shape lines ( 4710 - 1   a ,  4710 - 1   b ). The launch pad portion  4710 - 1   b  is connected to the metamaterial transmission line  4736 - 1  and may be of about 0.4 mm×3.2 mm. The launch pad portion  4710 - 1   a  is capacitively coupled to the cell patch  4701 - 1  and may be about 4.7 mm×1.5 mm. The cell patch  4701 - 1  can be connected to the via pad  4715 - 1  through a via  4705 - 1 . The via  4705 - 1  may be further connected to the cell patch  4701 - 1  on a first side of the substrate and connected to a via pad  4715 - 1  on the opposing side of the substrate. The via  4705 - 1  radius may be about 0.15 mm and the via center may be about 2.96 mm away from the top open end of the cell patch  4705 - 1 . The via pad  4715 - 1  may be about 0.6 mm×0.6 mm. The ground line  4725 - 1 , which may be about 0.2 mm×8.6 mm, can be connected to the via pad  4715 - 1  and to the microstrip GND  4715 .  
      The metamaterial transmission lines  4736 - 1  and  4736 - 2  shown in  FIG. 47B  may be realized by using a 2-cell CRLH structure. Each metamaterial transmission line ( 4736 - 1  and  4736 - 2 ) can have a right-handed (RH) and left-handed (LH) portion. Referring again to  FIG. 47A , the RH portion may be implemented by two identical sections of 50Ω microstrip lines ( 4735 - 1   a  and  4735 - 1   b ) and the LH portion is implemented by using chip capacitors ( 4730 - 1  and  4745 - 1 ) and shorted stubs  4740 - 1 . In this example, each microstrip section ( 4735 - 1   a  and  4735 - 1   b ) may be about 1.4 mm×2 mm. The two microstrip sections are connected to each other through three series capacitors ( 4745 - 1 ,  4730 - 1 ) in the order of 2C L , C L  and 2C L  where C L  may be about 1.6 pF. Two capacitor pads  4737  shown in  FIG. 47C  are placed in between the two microstrip sections  4735 - 1   a  and  4735 - 1   b  and used as the mounting base of the chip capacitors ( 4745 - 1  and  4730 - 1 ). The spacing between microstrip section ( 4735 - 1   a  or  4735 - 1   b ) and the adjacent capacitor pad  4737  may be 0.4 mm. The spacing between two capacitor pads  4737  may be 0.4 mm. The capacitor pads  4737  may be about 0.5 mm×0.6 mm. One side of two shorted stubs  4740 - 1  are attached at the center of the capacitor pads  4737  while the other side of the two shorted stubs  4740 - 1  is connected to via pads  4749 - 1 . The via pads  4749 - 1  may be connected to the microstrip GND  4715  through vias  4748 - 1 . The shorted stub  4740 - 1  may include three sections having the same width of about 0.2 mm and varying lengths of about 5 mm, 1.3 mm and 0.9 mm, respectively. The via pad  4749 - 1  may have a dimension of about 0.762 mm×0.762 mm. The vias  4748 - 1  is connected to the via pads  4749 - 1  on a first side of a substrate and to the microstrip GND  4715  on the opposing side of the substrate. The radius of the vias  4748 - 1  may be about 0.254 mm and may be centered with respect to the via pads  4749 - 1 .  
       FIG. 48  shows additional details of the MTM coupler  4800  of the multi-antenna system presented in this section. The MTM coupler  4800  has four ports that can be used as an input and output of the MTM coupler  4800 , respectively. In this example ports P 1   4845 - 1  and P 2   4845 - 2  can be used for inputs while ports P 3   4845 - 3  and P 4   4845 - 4  can be used as the outputs of the MTM coupler  4800 . Ports P 3   4845 - 3  and P 4   4845 - 4  can be connected to the inputs P 1 ′  4750 - 1  and P 2 ′  4750 - 2  of metamaterial antenna array  4700  shown in  FIG. 47A . The detailed descriptions of the MTM coupler  4800  is similar to the MTM coupler  4100  shown in  FIGS. 41A-41C .  
      The multi-antenna system in this section is simulated by using Ansoft HFSS.  FIG. 49  illustrates the return losses and coupling level between the two inputs of the multi-antenna system shown in  FIGS. 46A-46D  in which an improvement of the isolation between the two inputs is achieved as compared to the result shown in  FIG. 35 .  
       FIGS. 50A-50C  illustrates the radiation patterns of the multi-antenna system shown in  FIGS. 46A-46D  which show two opposite beam directions with respect to two inputs can occur. Such results generally indicate successful pattern diversity and low far-field envelope correlation of the multi-antenna system presented in this implementation.  
               TABLE 4                          Multi-Antenna, Directional Coupler System: Two-Element       Antenna Array with MTM Transmission Line Feed, Two-way       Directional Coupler using MTM Transmission Line (single band)                         Parameter   Description   Location               Multi-   Multi-antenna system includes an MTM           Antenna   Antenna Array and an MTM Coupler.       System       MTM   MTM Antenna array includes two MTM       Antenna   Antenna Elements with MTM Transmission       Array   Line feeds.       MTM   Each antenna element includes a MTM       Antenna   Cell coupled to the 50 Ω MTM       Element   Transmission Line via a Launch Pad.           Launch Pad is located on top of the           substrate.       Launch Pad   Two rectangular shape patches that   Top Layer           connect Cell Patch to the 50 Ω MTM           Transmission Line. There is a coupling           gap between the Launch Pad and the Cell           Patch.                             MTM Cell   Cell   Rectangular shape   Top Layer           Patch           Via   Cylindrical shape and   Top Layer               connects the Cell Patch   to Bottom               with the Via Pad.   Layer           Via Pad   Small square pad that   Bottom               connects the bottom part of   Layer               the Via to the GND Line.           GND Line   Connects the Via Pad to the   Bottom               microstrip GND.   Layer       MTM   Microstrip   Rectangular shape.   Top Layer       Transmission   Line   Characteristic impedance of       Line   Section   50 Ω.           Series   Chip capacitor (2*CL) which   Top Layer           Capacitors   connects one end of the               Microstrip Line Section and               one end of the Capacitor               Pad. Chip capacitor (CL)               which connects between two               Capacitor Pads.           Capacitor   Rectangular shape   Top Layer           Pad           Shorted   This stub includes three   Top Layer           Stub   Thin Microstrip Sections,   to Bottom               two Microstrip Bends, Via   Layer               Pad and a Via.           Thin   Rectangular shape.   Top           Microstrip       Layer           Section           Microstrip   Triangular shape of   Top Layer           Bend   microstrip junction to               connect two perpendicular               Thin Microstrip Section           Via Pad   Rectangular shape.   Top Layer           Via   Cylindrical shape.   Top Layer               Connecting Via Pad to   to Bottom               microstrip ground.   Layer                         MTM   MTM coupler includes an MTM Coupled           Coupler   Line, four Tapered Lines, and Four           Microstrip Bend       MTM   Two metamaterial transmission lines       Coupled   parallel to each other.       Line                                 Microstrip   Rectangular shape.   Top Layer           Line           Series   Chip capacitor (2*CL) which   Top Layer           Capacitor   connects one end of the               Microstrip Line and one end               of the Capacitor Pad. Chip               capacitor (CL) which               connects between two               Capacitor Pads in the               directional parallel to the               Microstrip Line.           Coupling   Chip capacitor (Cm) which   Top Layer           Capacitor   connects between two               Capacitor Pads in the               directional perpendicular to               the Microstrip Line.           Capacitor   Rectangular shape.   Top Layer           Pad           Shorted   Shorted stub includes a   Top Layer           Stub   Microstrip Stub, a Via Pad,               and a Via.           Microstrip   Rectangular shape.   Top Layer           Stub           Via Pad   Square shape.   Top Layer           Via   Cylindrical shape.   Top Layer               Connecting Via Pad to               microstrip ground.                         Tapered Line   Microstrip line with different line   Top Layer           width at both ends.       Microstrip   Triangular shape of microstrip junction   Top Layer       Bend   to connect two perpendicular microstrip           lines.                    
 III.A4 Two-Element Antenna Array with Two-way Directional Coupler Using MTM Transmission Line (USB Dongle Applications) 
 
      The multi-antenna system shown in  FIG. 31A  can be applied to the USB dongle applications.  FIGS. 51A-51D  illustrates another implementation of the multi-antenna system for USB applications. To realize multi-antenna system in a USB dongle, the available area of the multi-antenna system used in USB applications is generally smaller than the available area described in the previous implementations.  
      In another implementation of the multi-antenna system, a coplanar waveguide (CPW) MTM coupler can be used to improve the isolation between the two metamaterial antenna elements. To reduce the overall system size, the feed lines of the antennas are eliminated as illustrated in  FIG. 52A . Each element is described in detail in Table 5.  
      In another implementation, the multi-antenna system shown in  FIGS. 51A-51D  and  FIGS. 52A-52C  can be designed on a 1-mm FR4 substrate with dielectric constant of 4.4.  FIG. 52B  represents the same multi-antenna system shown in  FIG. 52A  and depicts specific elements. Referring to  FIGS. 52A-52C , the metamaterial antenna array may include two MTM antenna elements Ant 1  ( 5201 - 1 ,  5201 - 2 ) where the spacing between the inner edges of the antennas measures about 9.2 mm. Ant 1   5201 - 2 , for example, may be capacitively coupled through a coupling gap  5260  to one end of the L-shape launch pad  5205 . The other end of the L-shape launch pad  5205  is connected to the ports P 1 ′  5225 - 3  and P 2 ′  5225 - 4  which can be used as the outputs of the CPW MTM coupler or the inputs of the Ant 1  ( 5201 - 1  and  5201 - 2 ). A cell patch  5250  of the Ant 1   5201 - 2  may have a dimension of about 3.8 mm×7 mm and the dimension of the coupling gap may be about 3.8 mm×0.1016 mm. The L-shape launch pad  5205  may include a rectangular line, two 90° bends and a tapered line  5207  as shown in  FIGS. 52A-52C . The dimension of the rectangular line may be about 5.73 mm×0.6 mm. For the tapered line  5207 , the dimension may be about 3.27 mm in length and may have a first width of 0.6 mm on one side and a second width of 0.83 mm on the other side. The rectangular line of the launch pad  5205  is connected to tapered line  5207  through a first 90° bend while the tapered line  5207  is connected to the CPW MTM coupler through a second, larger 90° bend. The cell patch  5250  may be also connected to the CPW ground  5265  through a via  5203 , and an L-shape ground line  5270 . The via  5203  connects the cell patch  5250  on one side of the substrate and a via pad  5255  on the opposite side of the substrate. The radius of the via  5203  may be about 0.127 mm and may be centered at about 6.5 mm away from the CPW ground  5265  and 5.2016 mm from the open end portion of the cell patch  5250 . The via pad  5255  may have a dimension of about 0.8 mm×0.8 mm. The L-shape ground line  5270  may include two rectangular lines and a 90° bend. The first rectangular line is connected to the via pad  5255  and may be about 0.3 mm×1.8 mm while the second rectangular line is connected to the CPW ground  5265  and may have a dimension of about 0.3 mm×6.35 mm. The 90° bend located on both sides of connection may have a width of about 0.3 mm.  
      The CPW MTM coupler illustrated in  FIGS. 52A-52C , may include four ports where, in this implementation, ports P 1   5225 - 1  and P 2   5225 - 2  are used for RF inputs while the two outputs P 1 ′  5225 - 3  and P 2 ′  5225 - 4  are connected to the metamaterial antenna array ( 5201 - 1 ,  5201 - 2 ), respectively. A 50Ω CPW feed line  5240  includes two rectangular CPW sections and two 50Ω CPW bends  5130  and may have a dimension of about 0.83 mm×6.155 mm with 0.15 mm spacing to the CPW ground  5265 . Two connection sides of the 50Ω CPW bend  5130  may have a width of about 0.83 mm. The coupling portion of this coupler is realized by a MTM CPW coupled line  5215  where two CPW MTM transmission lines are placed in parallel to each other and are connected by a coupling capacitor Cm  5235 . The total length of the CPW MTM coupled line  5215  in this example may be about 4.4 mm, and the gap between two CPW MTM transmission lines may be about 1 mm. The chip capacitor C m    5235  (e.g., 0.4 pF) can be used to enhance the coupling between two MTM CPW transmission lines. Each MTM CPW transmission line may include two segments of CPW lines  5217 , a capacitor pads  5220 , two series capacitors  5245  (2*C L ) and one CPW shorted stub  5210 . The CPW segments can be identical in this CPW MTM coupler design and each section may have a dimension of about 0.83 mm×1.5 mm. The two CPW lines  5217  on one side can be connected by two series capacitors  5245  of 2C L  and a capacitor pad  5220 . The capacitor pad  5220  between the two CPW lines  5217  is used as a metal base to mount the series capacitors  5245 . In this example, 2C L  is realized by using a chip capacitor which may have a value of 1.5 pF. The spacing between the CPW lines  5217  and the capacitor pad  5220  may be about 0.4 mm. The capacitor pad  5220  may be about 0.6 mm×0.8 mm. The CPW shorted stub  5210  can be implemented by using a CPW stub where one side of the CPW stub is attached to the capacitor pad  5220  while the other side is connected directly to the CPW ground  5265 . In this example, the CPW shorted stub  5210  may have a dimension of about 0.15 mm×2.5 mm and has a gap to the CPW ground  5265  with a gap which may be about 0.225 mm.  
      The multi-antenna system shown in  FIGS. 52A-52C  is simulated by using Ansoft HFSS.  FIG. 53  shows the return losses and the coupling level between the two MTM antenna elements ( 5201 - 1 ,  5201 - 2 ) of  FIG. 52A  without the CPW MTM coupler.  FIG. 54  illustrates the return losses and the coupling level for the present implementation of the multi-antenna system shown in  FIGS. 52A-52C  which demonstrates significant improvement of the isolation by using the CPW MTM coupler.  FIGS. 55A-55C  illustrates the radiation patterns of the present implementation of multi-antenna system shown in  FIGS. 52A-52C  which show two opposite beam directions with respect to two RF inputs can occur. Such results generally indicate successful pattern diversity and low far-field envelope correlation of the multi-antenna system presented in this implementation.  
               TABLE 5                          Multi-Antenna, Directional Coupler System: Two-Element       Antenna Array, Two-way Directional Coupler using MTM       Transmission Line (USB Dongle Applications)                         Parameter   Description   Location               Multi-   Multi-antenna system includes an           Antenna   Metamaterial Antenna Array and an CPW       System   MTM Coupler.       Metamaterial   Antenna array includes two MTM Antenna       Antenna   Elements.       Array       MTM   Each antenna element includes an MTM       Antenna   Cell coupled to the 50 Ω MTM CPW Coupled       Element   Line via a Launch Pad. Launch Pad is           located on top of the substrate.       Launch Pad   L-shape. Launch pad includes one   Top Layer           rectangular line and one Tapered Line           and two 90° Bends.                                 Tapered   Microstrip line with   Top Layer           Line   different line widths at both               ends.           90° Bend   Triangular shape.   Top Layer       MTM Cell   Cell   Rectangular shape   Top Layer           Patch           Via   Cylindrical shape and   Top Layer               connects the Cell Patch with   to Bottom               the GND Pad.   Layer           GND Pad   Small square pad that   Bottom               connects the bottom part of   Layer               the Via to the GND Line.           GND Line   Connects the GND Pad to the   Bottom               main CPW GND   Layer                         CPW MTM   CPW MTM Coupler includes a MTM CPW           Coupler   Coupled Line, two CPW Feed Lines, and           four CPW Bends.       MTM CPW   Two MTM CPW Transmission Lines parallel       Coupled   to each other. Each MTM CPW Transmission       Line   Line includes two CPW Segments, Series           Capacitor, Capacitor Pad, and CPW           Shorted Stub.                                 CPW   Rectangular shape.   Top Layer           Segment           Series   Chip capacitor (2*CL) which   Top Layer           Capacitor   connects one end of the CPW               Segment and one end of the               Capacitor Pad.           Coupling   Chip capacitor (Cm) which   Top Layer           Capacitor   connects between two               Capacitor Pads in the               directional perpendicular to               the CPW Segment.           Capacitor   Rectangular shape.   Top Layer           Pad           CPW   CPW line shorted to the CPW   Top Layer           Shorted   GND.           Stub                         CPW Feed   L-shape with CPW Bend at the joint and   Top Layer       Line   at the connection to the MTM CPW Coupled           Line.       CPW Bend   Triangular shape of CPW junction to   Top Layer           connect two perpendicular CPW lines.                  
 
 IV. Multi-Antenna, Directional Coupler System: Full Duplex Communication Support 
 
       FIG. 56A  illustrates a multi-antenna system for a time division duplex application. The antennas are used to either transmit or receive at different time instants. In this example, one antenna is used to transmit a signal to user i while the other antenna is used to receive a signal from user j as illustrated in  FIG. 56B . The Tx and Rx signals can also target a single user in a multipath environment where both signals bounce off scattering objects opening two different paths between the multi-antenna system and an end user. As illustrated in  FIG. 56B , the transmit signal is coupled with the received signal at the transmit antenna port. But since the received signal power is much small than the transmit signal power, which is further reduced by the coupling factor, it has minimal impact on the transmit signal quality. Similarly, the signal received on the receiver port may include three components: 1) signal received from the Rx antenna, 2) transmit signal coupled to the receive port, and 3) transmit signal coupled through the air. In the case of the present implementation of the multi-antenna system, the two coupling coefficients C 1  and C 2  are equal in magnitude and opposite in phase. As a result at the receiver port, all the transmitter power is cancelled and only the signal seen by the receive antenna is received at the port. In comparison, other technologies generally have high isolation required between Tx and Rx antennas and, thus, tend to make it difficult to achieve this solution. Such multi-user solution can be used on the client side, access-point or base-station, or on both allowing unique deployment of wireless networks.  
      In another application, the multi-antenna system in  FIG. 56B  can be used to eliminate the Tx/Rx switch in a time-division duplex system. As explained above, the transmitted signal may be coupled to the transmit antenna and the receive signal at the receive antenna may be coupled to the receive port resulting in minimal mutual coupling between the two paths. As a result, the need for transmit/receive switch can be eliminated.  
      V. Dualband Multi-Antenna System: Two-Element Antenna Array with 2-Way Directional Coupler  
      A microwave directional coupler can be used to decouple two coupled antenna elements. This approach can be applied also to a multiband antenna system.  
       FIG. 57A  and  FIG. 57B  illustrates a configuration of a dual-band multi-antenna system  5700 -A and  5700 -B, respectively. Four signal transmission paths are denoted as path 1   5701 - 1 , path 2   5701 - 2 , path 3   5701 - 3  and path 4   5701 - 4 . These paths are characterized by coupling magnitudes C 1 , C 2 , C 3  and C 4  and phases θ 1 , θ 2 , θ 3  and θ 4  at the first frequency f1, and C 1 ′, C 2 ′, C 3 ′ and C 4 ′ and phases θ 1 ′, θ 2 ′, θ 3 ′ and θ 4 ′ at the second frequency f2, respectively. Unlike the conventional antenna system where each antenna element is placed at ˜0.5λ 0  where λ 0  is free space wavelength away from the adjacent antenna elements to minimize the isolation, the spacing d  5703  between two antenna elements ( 5705 ,  5707 ) in this dual-band multi-antenna system  5700  can be much smaller, e.g., from 0.1λ 0  up to 0.25λ 0 .  
      Two examples are considered below. The first case, the antenna array has strong coupling (e.g., larger than −10 dB) at both frequencies f1 and f2. The second case, the antenna array has strong coupling at f1 but weak coupling (e.g., less than −10 dB) at f2 where f2&gt;f1.  
     Example 1  
     Antenna Array has Strong Coupling at f1 and f2  
      The conditions to decouple two antenna elements are expressed as:  
       at   ⁢           ⁢   f   ⁢           ⁢   1   ⁢           ⁢     {                 θ   2     +     θ   3     +     θ   4     -     θ   1       =       -   180     ⁢   °                     ⁢     Eq   .           ⁢     (     14   ⁢           ⁢   a     )                     C   1     =       C   2     ·       C   ⁢             3     ·     C   4                       ⁢     Eq   .           ⁢     (     14   ⁢           ⁢   b     )               ⁢     
     ⁢   at   ⁢           ⁢   f   ⁢           ⁢   2   ⁢           ⁢     {               θ   2   ′     +     θ   3   ′     +     θ   4   ′     -     θ   1   ′       =       -   180     ⁢   °                     ⁢     Eq   .           ⁢     (     14   ⁢           ⁢   c     )                     C   1     =       C   2   ′     ·     C   3   ′     ·     C   4   ′                       ⁢     Eq   .           ⁢     (     14   ⁢           ⁢   d     )                         
 
      By introducing the following relationships of a symmetric directional coupler: 
 
θ 2 =θ 4   Eq. (15a) 
 
θ 1 ≈θ 2 +90°  Eq. (15b) 
 
θ 2 ′=θ 4 ′  Eq. (15c) 
 
θ 1 ′≈θ 2 ′+90°  Eq. (15d) 
 
      we get the following relationships between the phases at f1 and the phases at f2:  
               θ   2     ≈         -   90     ⁢   °     -     θ   3               Eq   .           ⁢     (     16   ⁢           ⁢   a     )                     θ   2   ′     ≈         -   90     ⁢   °     -     θ   3   ′         ⁢     
     ⁢   And           Eq   .           ⁢     (     16   ⁢           ⁢   b     )                   θ   2   ′     =         f   ⁢           ⁢   2       f   ⁢           ⁢   1       ·     θ   2   ′               Eq   .           ⁢     (     16   ⁢           ⁢   c     )               
 
      In addition, using the assumptions of C 2 =C 4 ≈1 and C 2 ′=C 4 ′≈1 that are applicable to most low loss directional couplers, we obtain the following relationships: 
 
C 1 ≈C 3   Eq. (17a) 
 
C 1 ′≈C 3 ′  Eq. (17b) 
 
      It should be noted that C 1  has to be smaller than C 3 . The zero coupling can be obtained at two frequencies f1 and f2 if the Eq. (16a)-(16c) and Eq. (17a)-(17b) are simultaneously satisfied.  
     Example 2  
     Antenna Array has Strong Coupling at f1 and Weak Coupling at f2 While f2&gt;f1  
      If C 3 ′ is small, that is, the isolation between two antenna elements is sufficient, the decoupling circuit may not be necessary. Therefore, the conditions to achieve the dual-band antenna system with high isolation using the coupler network are expressed as follows:  
       at   ⁢           ⁢   f   ⁢           ⁢   1   ⁢           ⁢     {               θ   2     +     θ   3     +     θ   4     -     θ   1       =       -   180     ⁢   °                     ⁢     Eq   .           ⁢     (     18   ⁢           ⁢   a     )                     C   1     =       C   2     ·       C   ⁢             3     ·     C   4                       ⁢     Eq   .           ⁢     (     18   ⁢           ⁢   b     )                     
 
 Based pm the following relationships of a symmetric directional coupler; 
 
θ 2 =θ 4   Eq. (19a) 
 
θ 1 ≈θ 2 +90°  Eq. (19b) 
 
 the following relationship can be obtained: 
 
θ 2 =−90°−θ 3   Eq. (20) 
 
      In addition, assuming that C 2 =C 4 ≈1 and C 3 ′ is weak, the following relationships can be derived: 
 
C 1 ≈C 3   Eq. (21a) 
 
C 3 ′&lt;&lt;1  Eq. (21b) 
 
 where C 1  is smaller than C 3 . The high isolation between two antenna elements can be achieved if Eq. (20) and Eq. (21a)-(21b) are satisfied. 
 
      The directional coupler shown in  FIG. 57  can be implemented by using a conventional transmission line technology such as microstrip line and coplanar waveguide (CPW) or by using MTM technology. The MTM technology has several advantages over the conventional transmission line technology. First, the MTM coupler can achieve broader bandwidth. Second, the MTM coupler can provide up to 0 dB coupling whereas the conventional coupler can only provide up to around −8 dB coupling. Third, the MTM coupler can be made to occupy smaller space.  
      V.A1. Dualband Two-Element Antenna Array with 2-Way Directional Coupler Using Microwave Coupled Line—Condition: f2≠2×f1, f2&gt;f1, Strong Coupling at f1 and f2  
      In another embodiment of a multi-antenna system, a dual-band multi-antenna system using the MTM technology is shown in  FIG. 58A-58C . The present implementation of the dualband multi-antenna system may include a dualband two-element metamaterial antenna array and a conventional microwave directional coupler. Each element is described in detail in Table 6.  
               TABLE 6                          Two-Element Antenna Array, 2-Way Directional Coupler       using Microwave Coupled Line - Condition: f2 ≠ 2xf1, f2 &gt; f1,       strong coupling at f1 and f2 (Dualband)                         Elements   Description   Location               Multi-   Dualband Multi-antenna system comprises           Antenna   a Dualband MTM Antenna Array and a       System   Microwave Directional Coupler.       Dualband   Antenna array comprises two MTM Antenna       MTM   Elements.       Antenna       Array       MTM   MTM antenna element comprises an MTM       Antenna   Cell and a Launch Pad.       Element       Launch Pad   Each Launch Pad comprises two   Top Layer           rectangular shape patches, one of which           connects to the Cell Patch and the           other connects to the 50 Ω CPW feed           line. There is a coupling gap between           the Launch Pad and the MTM Cell.                             MTM Cell   Cell   Rectangular shape.   Top Layer           Patch           Via   Cylindrical shape and   Top Layer               connects the Cell Patch with   to Bottom               the Via Pad.   Layer           Via Pad   Small square pad that   Bottom               connects the bottom part of   Layer               the Via to the GND Line.           GND Line   Connects the Via Pad to the   Bottom               main GND.   Layer                         Microwave   Directional coupler comprises a           Directional   Microstrip Coupled Line, four Tapered       Coupler   Lines, four Microstrip Bend and four           CPW lines.       Microstrip   Two parallel microstrip lines with a   Top Layer       Coupled   coupling gap in between.       Line       Tapered   Microstrip line with different line   Top Layer       Line   width at both ends.       Microstrip   Triangular shape of microstrip junction   Top Layer       Bend   to connect two perpendicular microstrip           lines.                  
 
      As a specific example, the dualband multi-antenna system shown in  FIGS. 58A-58C  may be implemented on a 0.787 mm FR-4 substrate having a dielectric constant of 4.4. The metamaterial antenna array includes two metamaterial antenna elements. The metamaterial antenna elements, in this example, are connected to the 50Ω CPW feed line  5825  having a dimension of about 1.4 mm×20 mm with a gap to the CPW side ground  5859  of about 0.83 mm. The spacing between two antenna elements may be about 13 mm from the inner edges of the antenna elements. One side of the CPW feed lines  5825  is directly connected to the launch pads  5820  and the other side may be connected to the outputs of the microwave directional coupler  5805 . In this example, each launch pad  5820  may include two rectangular shape patches. The first rectangular patch which is connected to the CPW feed line  5825  and may have a dimension of about 0.4 mm×3.2 mm, and the second rectangular patch is capacitively coupled to the cell patch  5801  which may have a dimension of about 4.7 mm×1.5 mm. The cell patch  5801  is coupled to the launch pad  5820  through a coupling gap  5823  of about 0.16 mm and is shorted to the main ground  5840  through a via  5855 , via pad  5850  and a ground line  5845 . The dimension of the cell patch  5801 , as shown in this example, may be about 4.7 mm×7 mm. The via  5855  can connect the cell patch  5801  on top side of the dielectric substrate  5830  and to the via pad  5850  on the bottom side of the dielectric substrate  5830 . The radius of the via  5855  may be about 0.15 mm and its center may be located at about 2.96 mm from the top open end of the cell patch  5801 . The dimension of the via pad  5850  may be about 0.6 mm×0.6 mm and is connected to the main ground  5840  through a ground line  5845 . The ground line  5845  may have a dimension of about 0.2 mm×8.6 mm.  
       FIG. 58B  illustrates the top view of the top layer  5815  depicted in  FIG. 58A  and  FIG. 58C  illustrates the top view of the bottom layer  5835  also depicted in  FIG. 58A . Elements shown in  FIGS. 58B-58C  which are also represented in  FIG. 58A  include cell patch  5861 , launch pad  5863 , CPW feed line  5865 , CPW Side Ground  5869 , CPW Line  5873 , Via Pad  5877 , GND Line  5879 , and Main Ground  5881 . Additional elements depicted in  FIG. 58B  and previously mentioned include tapered line  5867 , microstrip bend  5871 , and microstrip coupled line  5875 .  
      The MTM antenna array in  FIGS. 58A-58C  without the microwave directional coupler  5805  is simulated by using Ansoft HFSS. The simulation results are shown in  FIG. 59  where the coupling and the return losses are plotted as a function of frequency.  FIG. 59  shows that the designs of the antenna array and the directional coupler described above make the device to have a strong coupling between two adjacent antennas at two different frequencies f1 and f2 that are not harmonic frequencies to each other. In this example, the metamaterial antenna array operates at two frequencies, f1=2.33 GHz and f2=5.1 GHz-6 GHz, and the coupling is about −7.4 dB and −8.1 dB at f1 and f2, respectively. Since the couplings at these two frequencies are strong (more than −10 dB), the conditions mentioned in Example 1 in Section V are considered to design the microwave directional coupler  5805 .  
      The expanded top view of the microwave directional coupler  5805  in  FIG. 58A  is shown in  FIG. 60A , where in this example Port 1   6001  and Port 3   6003  are used for RF inputs and Port 2   6002  and Port 4   6004  are the outputs of this microwave directional coupler. Port 2   6002  and Port 4   6004  are connected to the inputs of the metamaterial antenna array shown in  FIGS. 58A-58C . The dimensions of the CPW lines  6025  for the two coupler inputs may be of 1.48 mm×5 mm, and the gap to the CPW side ground  6005  may be about 0.83 mm. The dimensions of the CPW lines  6020  for the two coupler outputs may be of 1.4 mm×3.65 mm, and the gap to the CPW side ground  6005  may be 0.83 mm. Both input and output CPW lines ( 6025 ,  6020 ) can have characteristic impedance of around 50Ω. The coupling portion of this coupler can be realized by using a microstrip coupled line  6030  where the length of the coupled line, the width of the coupled line, and the coupling gap may be 12 mm, 0.4 mm and 0.1 mm, respectively. The four ends of the microstrip coupled line  6030  can be connected to the four CPW lines ( 6020 ,  6025 ) through the four microstrip tapered lines and the four microstrip bends  6029  for the impedance matching purpose. In this implementation, the length of the microstrip tapered lines  6027  is connected to the RF inputs (Port 1   6001 , Port 3   6003 ) and may be about 8.8 mm. The widths for the microstrip tapered line  6027  may be about 1.48 mm at one end portion and about 0.4 mm at the other end portion. The microstrip tapered lines  6027  are connected to the coupler output ports Port 2   6002  and Port 4   6004  and their lengths may be about 5.35 mm. The widths for the microstrip tapered lines  6027  may be about 1.4 mm in one end portion and about 0.4 mm in the other end portion. The microwave directional coupler, in this example, can be simulated by using Ansoft HFSS.  FIG. 60B  illustrates the return loss, insertion loss and coupling for the present implementation of the microwave directional coupler shown in  FIG. 60A  with signal input at Port 1   6001 . The simulated results shown in  FIG. 60B  demonstrates good impedance matching and sufficient coupling between port 1   6001  and port 3   6003  over a frequency range from about 1.8 GHz to 5.3 GHz.  
      The dualband multi-antenna system of  FIGS. 58A-58C  is simulated by using Ansoft HFSS.  FIG. 61  shows the return losses and coupling level between the two metamaterial antenna array elements in  FIGS. 58A-58C . The results of  FIG. 61  demonstrates that the isolation between the two antenna elements can be significantly improved in comparison to the case without the microwave directional coupler ( FIG. 59 ) while still maintaining a good return loss at the two frequencies, 2.33 GHz and 4.95 GHz. At these two frequencies, Eq. (16a-16c) and Eq. (17a-17b) are satisfied.  
      V.A2. Dualband Two-Element Antenna Array with 2-Way Directional Coupler Using Microwave Coupled Line—Condition: f2=2×f1, f2&gt;f1, Strong Coupling at f1 and Weak Coupling at f2  
      Another dual-band multi-antenna system can be designed to include a two-element metamaterial antenna array and a conventional microwave directional coupler. A detailed description of each element presented for the dual-band multi-antenna system is described in Table 7 and  FIG. 62  and  FIGS. 63A-63B .  FIG. 63A  illustrates the top layer  6220  of  FIG. 62 , and  FIG. 63B  illustrates the bottom layer  6330  of  FIG. 62 .  
               TABLE 7                          Two-Element Antenna Array, 2-Way Directional Coupler       using Microwave Coupled Line - Condition: f2 = 2xf1, f2 &gt; f1,       strong coupling at f1 and weak coupling at f2 (Dualband)                         Elements   Description   Location               Dualband   Dualband multi-antenna system comprises           Multi-   a Dualband MTM Antenna Array and a       Antenna   Microwave Directional Coupler.       System       Dualband   Antenna array comprises two MTM Antenna       MTM Antenna   Elements.       Array       MTM Antenna   Each antenna element comprises an MTM       Element   Cell and a Launch Pad.       Launch Pad   Each Launch Pad comprises two   Top Layer           rectangular shape patches, one of which           connects to the MTM Cell and the other           one connects to the 50 Ω CPW feed line.           There is a coupling gap between the           Launch Pad and the Cell Patch.                             MTM Cell   Cell   Rectangular shape   Top Layer           Patch           Via   Cylindrical shape and   Top Layer               connects the Cell Patch with   to Bottom               the Via Pad.   Layer           Via Pad   Small square pad that   Bottom               connects the bottom part of   Layer               the Via to the GND Line.           GND Line   L shaped line that connects   Bottom               the Via Pad to the main GND.   Layer                         Microwave   Directional coupler comprises a   Top layer       Directional   Microstrip Coupled Line.       Coupler       Microstrip   Two microstrip line parallel with each   Top layer       Coupled   other with a gap in between.       Line                  
 
      The metamaterial antenna array shown in  FIG. 62  and  FIGS. 63A-63B  can be implemented on a 1-mm FR-4 substrate with dielectric constant of 4.4. Each of the antenna element, in this example, can be fed by a 50Ω CPW feed line  6210  and has a dimension of about 0.83 mm×22.88 mm. The length of the CPW feed line  6210  can be selected to satisfy the phase requirement. The spacing between the inner edges of two antenna elements may be about 8.4 mm. One end portion of the CPW feed lines  6210  can be directly connected to the launch pads  6205  and the other end portion can be connected to the outputs of the microwave directional coupler, as described in the next section or to the inputs of the metamaterial antenna elements. Each of the launch pads  6205  may include two rectangular shape patches. The first rectangular patch is connected to the CPW feed line  6210  and may have a dimension of about 0.6 mm×4.1 mm. The second rectangular patch is capacitively coupled to the cell patch  6201  and may have a dimension of about 1 mm×4.4 mm. The cell patch  6201  can be coupled to the launch pad  6205  through a coupling gap  6208  which may be about 0.1524 mm and can be shorted to a ground  6255  through a via  6240 , via pad  6245  and ground line  6235 . The dimension of the cell patch  6201 , in this example, may be about 4.4 mm×7 mm. The via  6240  is connected to the cell patch  6201  on the top side of a dielectric substrate  6225  and to a via pad  6245  on the bottom side of the dielectric substrate  6225 . The radius of the via  6240  may be about 0.127 mm, and its center may be located at about 3.3524 mm from the open end portion of the cell patch  6201 . The via pad  6245  is connected to the ground  6255  through an L-shape ground line  6235  and may have a dimension of about 0.8 mm×0.8 mm. The ground line  6235  includes a first arm which is connected to the via pad  6245  and may have a dimension of about 0.3 mm×4.1 mm, and a second arm that is connected to the ground  6255  and may have a dimension of about 0.3 mm×6.35 mm.  
      The metamaterial antenna array can be simulated by using Ansoft HFSS, and the results are shown in  FIGS. 64A-64B . The results of these figures show that the metamaterial antenna array can operate at two different frequencies, f1=2.5 GHz and f2=5.0 GHz which is a second harmonic frequency of f1. The designs of the antenna array and the directional coupler are selected to have a strong coupling between two adjacent antennas at f1 and a weak coupling at f1. In the example in  FIG. 64A , the coupling between the two antennas is −6.47 dB and −15.67 dB at f1 and f2, respectively. Since the coupling at f2 is weak, the conditions mentioned in example 2 in Section V may be considered to design the microwave directional coupler.  
      The structure of the microwave directional coupler which can be implemented using microstrip coupled lines is shown in  FIG. 65A . In this example, the microwave directional coupler can be designed on a 1 mm FR-4 substrate having dielectric constant of 4.4. As shown in  FIG. 65A , the width w  6515  of the microstrip coupled line measures about 1.3162 mm, the length L  6510  measures about 16.7941 mm, and the coupling gap s  6505  measures about 0.2843 mm.  
      The microwave directional coupler can have four ports where ports P 1   6501 - 1  and P 3   6501 - 3  may be used for RF inputs, and ports P 2   6501 - 2  and P 4   6501 - 4  may be used as the outputs of the coupler, as shown in  FIG. 65A . Ports P 2   6501 - 2  and P 4   6501 - 4  is connected to the metamaterial antenna array as shown in  FIG. 62  and  FIGS. 63A-63B . From  FIG. 64B , the phase of 0° at 2.5 GHz may be obtained between P 1 ′  6215 - 1  and P 2 ′  6215 - 2  of  FIG. 62 . Thus, by using Eq. (20), the phase delay θ 2  from p 1   6501 - 1  to p 2   6501 - 2  in  FIG. 65A  may be found to be −90° at 2.5 GHz, and the coupling level |S 31 | may be defined as:  
                 C   ⁢           ⁢   3     =            S   ⁢           ⁢   31          =       j   ⁢           ⁢     k   ·     tan   ⁡     (     θ   2     )                 1   -     k   2         +     j   ·     tan   ⁡     (     θ   2     )                 ⁢     
     ⁢   where   ⁢     
     ⁢     k   =           Z     0   ⁢           ⁢   e       -     Z     0   ⁢           ⁢   o             Z     0   ⁢           ⁢   e       +     Z     0   ⁢           ⁢   o           ⁢           ⁢   and   ⁢           ⁢         Z     0   ⁢           ⁢   e       ·     Z     0   ⁢           ⁢   o                       Eq   .           ⁢     (   22   )               
 
      In Eq. (22), Z 0 , Z 0e , and Z 0o  are the characteristic impedance, even mode impedance and odd mode impedance, respectively, of the microstrip coupled lines shown in  FIG. 65A . The microwave directional coupler in this example, may be designed to have a characteristic impedance of 50Ω (Z 0 ) and a coupling (20 log |S 31 |) of −10 dB at 2.5 GHz. The maximum coupling can occur at θ 2 =−n·90° where n=1, 3, 5, 7 . . . . In this implementation, θ 2 =−90° and the maximum coupling can occur at 2.5 GHz, while the minimum coupling may occur at 5 GHz. Thus, equations Eq. (21a)-(21b) may be satisfied.  FIG. 65B  illustrates the simulated return loss, insertion loss, and coupling of the microwave directional coupler shown in  FIG. 65A  with input signal at P 1   6501 - 1 . Referring to  FIG. 65B , the microwave directional coupler can be matched well to 50Ω over a frequency range from 1 GHz to 6 GHz and may have a coupling of about −10 dB at 2.5 GHz and about −33 dB coupling at 5 GHz.  
       FIG. 66A  illustrates an example in which the metamaterial antenna array shown in  FIG. 62  and  FIGS. 63A-63B  is connected to the outputs (P 2   6501 - 2 , P 4   6501 - 4 ) of the microwave directional coupler in  FIG. 65A . In this implementation, the length L  6601  of the microstrip coupled line, the width w  6610  of the microstrip coupled line and the coupling gap s  6605  may be set to about 14.44 mm, 1.12 mm, and 0.23 mm, respectively. The simulation results for the dualband multi-antenna system of  FIG. 66A  are illustrated in  FIG. 66B . From these figures, an adequate return loss at 2.5 GHz and 5 GHz may be obtained while the isolations at these two frequencies can be less than about −10 dB.  
      V.A3. Dualband Two-Element Antenna Array with 2-Way Directional Coupler Using MTM Transmission Line—Condition: f2≠2×f1, f2&gt;f1, Strong Coupling at f1 and Weak Coupling at f2  
      The use of a conventional microwave directional coupler to improve the isolation between two antenna array elements at two frequencies has been demonstrated in the previous sections. In previous case, design of the coupler may be easier since only the requirement on the phase at f1 had to be satisfied. However, when using the conventional microwave directional coupler, the second frequency f2 has to be the even multiple of the first frequency f1 due to linearity of the transmission line propagation constant. Therefore, in order to design a dual-band multi-antenna system with flexibility, a different type of directional coupler may be required. In this case, an MTM coupler may be used to decouple two coupled metamaterial antenna array elements with f2≠2×f1. In another implementation of a multi-antenna system, a dual-band multi-antenna system may include a two-element metamaterial antenna array and an MTM coupler. A detailed description of each element is presented in Table 8.  
               TABLE 8                          Multi-Antenna, Directional Coupler System: Two-Element       Antenna Array, 2-Way Directional Coupler using MTM       Transmission Line - Condition: f2 ≠ 2xf1, f2 &gt; f 1,       strong coupling at f1 and weak coupling at f2 (Dualband)                         Parameter   Description   Location               Dualband   Multi-antenna system comprises an MTM           Multi-   Antenna Array and a MTM Coupler.       Antenna       System       MTM Antenna   Antenna array comprises two MTM Antenna       Array   Elements.       MTM Antenna   Each antenna element comprises an MTM       Element   Cell and a Launch Pad.       Launch Pad   Each Launch Pad comprises two   Top Layer           rectangular shape patches, one of which           connects to the Cell Patch and the           other one connects to the 50 Ω CPW feed           line. There is a coupling gap between           the Launch Pad and the Cell Patch.                             MTM Cell   Cell   Rectangular shape   Top Layer           Patch           Via   Cylindrical shape and   Top Layer               connects the Cell Patch with   to Bottom               the Via Pad.   Layer           Via Pad   Small square pad that   Bottom               connects the bottom part of   Layer               the Via to the GND Line.           GND Line   L shaped line that connects   Bottom               the Via Pad to the main GND.   Layer                         MTM Coupler   MTM Coupler comprises two MTM               Transmission Lines in parallel to each           other with Mutual Coupling L-C Set in           between.       MTM   MTM transmission line comprises N Unit       Transmission   Cells cascading periodically along the       Line   direction of wave propagation.       Unit Cell   Each Unit cell comprises three sets of           inductor and capacitor combination           which include one Series L-C Set, one           Shunt L-C Set, and one Series C-L Set.                             Series   Series L-C set comprises one           L-C Set   series inductor and one               series capacitor in order.               The free end of the capacitor               connects to the Shunt L-C               Set.           Shunt   Shunt L-C set comprises one           L-C Set   shunt capacitor and one               series inductor.           Series   Series L-C set comprises one           C-L Set   series capacitor and one               series inductor in order. The               free end of the capacitor               connects to the Shunt L-C               Set.                     Mutual   Mutual coupling includes a mutual       Coupling L-C   inductance (L m ) and mutual capacitance       Set   (C m )                  
 
      The structure of the dual-band metamaterial antenna array can be the same as that of the dual-band metamaterial antenna array shown in  FIG. 62  and  FIGS. 63A-63B , except that some dimensions are different, and is implemented also on a 1 mm FR-4 substrate having a dielectric constant of 4.4.  
      The above MTM antenna array in Table 8 is simulated by using Ansoft HFSS, and the results are shown in  FIGS. 67A-67B . The results from these figures illustrate that the MTM antenna array described in this section may operate at two frequencies, f1=2.7 GHz and f2=5.0 GHz, and the coupling is about −6.27 dB and −15.63 dB at f1 and f2, respectively. Since the coupling at f2 is weak, the conditions mentioned in example 2 in Section V are considered to design the MTM directional coupler.  
      A metamaterial transmission line is an artificial transmission line structure and can be implemented by, for example, cascading N unit cells  6805  periodically. As shown in  FIG. 68A , the equivalent circuit model of a metamaterial unit cell  6805  comprises series capacitance (C L ), series inductance (L R ), shunt capacitance (C R ), and shunt inductance (L L ). In order to have symmetric response from the metamaterial transmission line, the symmetric unit cell  6815  depicted in  FIG. 68B  is used in this implementation. See Caloz and Itoh, “Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications,” John Wiley &amp; Sons (2006) for details in the equivalent circuit models. In  FIG. 68B , the series capacitance and inductance are divided into two branches where one branch is on the left hand side of the shunt elements and the other branch is on the right hand side of the shunt element. In order to mimic the unit cell circuit model drawn in  FIG. 68A , the series capacitance C L  and series inductance L R  are chosen to be 2C L  and L R /2, respectively, in each branch. In this implementation, the MTM coupler may be realized by coupling two metamaterial transmission lines in parallel.  
       FIG. 69  shows the equivalent circuit model of the MTM coupler. The coupling between the two metamaterial transmission lines is represented by using mutual inductance (L m ) and mutual capacitance (C m ) in the circuit model. In this example, port 1   6905 - 1  and port 3   6905 - 3  are used as the inputs, and port 2   6905 - 2  and port 4   6905 - 4  are used as the outputs of the MTM coupler which are to be connected to the inputs of the metamaterial antenna array elements.  
      In general, the propagation constant of a metamaterial transmission line is dispersive and has nonlinear response to the frequency. See, for example, Caloz and Itoh, “Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications,” John Wiley &amp; Sons (2006). Owing to this property, it may be possible to obtain maximum coupling and zero coupling at f1 and f2, respectively by using an MTM coupler, where f2 does not have to be even multiple of f1. Based on the simulation results for the metamaterial antenna array shown in  FIG. 67A , the MTM coupler may be designed to have maximum coupling at 2.7 GHz and zero coupling at 5 GHz. In this implementation, L L =7.5 nH, C L =3 pF, L R =1.249nH, C R =0.4996 pF, L m =0.2309 nH, and C m =0.11 pF are obtained. The number of unit cells may be chosen to be 5 to achieve sufficient coupling level.  FIG. 70  illustrates the return loss, insertion loss, and coupling of the MTM coupler represented by the equivalent circuit model in  FIG. 69 . From  FIG. 70 , the MTM coupler can be matched to 50Ω at both frequencies, 2.7 GHz and 5 GHz. The maximum coupling of −8.038 dB can be obtained at about 2.94 GHz, and about −33.29 dB coupling can be obtained at about 5 GHz.  
      The dual-band multi-antenna system can be constructed by connecting the outputs of the MTM coupler (port 2   6905 - 2  and port 4   6905 - 4 ) in  FIG. 69  directly to the two inputs of the metamaterial antenna array, which is similar in structure to the metamaterial antenna array in  FIG. 62  and  FIGS. 63A-63B .  FIG. 71  shows the simulation results of the return losses and insertion loss of the dual-band multi-antenna system described in this section. Sufficient isolations of about −19.82 dB and −18.64 dB between two elements of the metamaterial antenna array can be obtained at about 2.82 GHz and 5.08 GHz, respectively, while two antennas can be still matched to 50Ω at these two frequencies.  
      V.A4. Dualband Two-Element Antenna Array with 2-Way Vertical Directional Coupler—Condition: f2≠2×f1, f2&gt;f1, Strong Coupling at f1 and Weak Coupling at f2  
      To reduce the size of the whole system mentioned in the previous section, the microwave directional coupler in this section can be changed. Instead of using the microstrip coupled line for coupling, a coupled strip line may be used as the coupling portion. In this implementation, the dual-band multi-antenna system may include a two element metamaterial antenna array and a microwave vertical directional coupler. A detailed description of each element is described in Table 9.  
               TABLE 9                          Multi-Antenna, Directional Coupler System: Two-Element       Antenna Array, 2-Way Vertical Directional Coupler - Condition:       f2 ≠ 2xf1, f2 &gt; f1, strong coupling at f1 and weak coupling at f2       (Dualband)                         Elements   Description   Location               Dualband   Dualband multi-antenna system           Multi-   comprises an MTM Antenna Array and a       Antenna   microwave Vertical Directional       System   Coupler.       MTM Antenna   Antenna array comprises two MTM       Array   Antenna Elements and two 50 Ω CPW           Antenna Feed Lines.       MTM Antenna   Each antenna element comprises an       Element   MTM Cell and a Launch Pad.       Launch Pad   Each Launch Pad comprises two   Layer 1           rectangular shape patches, one of           which connects to the Cell Patch and           the other one connects to the 50 Ω           CPW feed line. There is a coupling           gap between the Launch Pad and the           Cell Patch.                             MTM Cell   Cell   Rectangular shape   Layer 1           Patch           Via   Cylindrical shape and   Layer 1 to               connects the Cell Patch   Layer 4               with the Via Pad.           Via Pad   Small square pad that   Layer 4               connects the bottom part               of the Via to the GND               Line.           GND   L shaped line that   Layer 4           Line   connects the Via Pad to               the main GND.                         50 Ω CPW   50 Ω CPW Antenna Feed Lines are on   Layer 1 and       Antenna   top and bottom of the substrate and   Layer 4       Feed Line   they are connected through vias.       Vertical   Vertical Directional Coupler       Directional   comprises four 50 Ω CPW Coupler Feed       Coupler   Lines, four Via Pads and one Coupled           Strip Line.       50 Ω CPW   Two 50 Ω CPW Coupler Feed Lines are   Layer 1 and       Coupler   on Layer 1 and connected to the via   Layer 4       Feed Line   pads on Layer 2 through vias. Another           two 50 Ω CPW Feed Lines are on Layer 4           and connected to the via pads on           Layer 3 through vias.       Via Pad   Small square pad that connects one   Layer 2 and           side of the via to one end of   Layer 3           Coupled Strip Line.       Coupled   Two strip line on top of each other   Layer 2 and       Strip Line   with a substrate layer in between.   Layer 3                  
 
       FIGS. 72A-72E  and  FIG. 73  illustrates a structure of the dual-band metamaterial antenna array. The metamaterial antenna array may be implemented on a 0.787 mm FR-4 substrate having a dielectric constant of 4.4. The space between the inner edges of the two antenna elements may be about 8.4 mm. Each metamaterial antenna can be fed by a 50Ω CPW feed lines  7204 ,  7215 . In  FIG. 72A , one end portion of the CPW feed line  7204  is connected directly to a launch pad  7202 - 1 , and the other end portion is connected to another CPW feed line  7215  on the other side of the substrate through a via  7205 . In  FIG. 72D , one end portion of the CPW feed line  7215  is directly connected to a launch pad  7202 - 2 , and the other end portion is connected to another CPW feed line  7204  on the other side of the substrate through via  7205 .  
      In this implementation, each launch pad ( 7202 - 1 ,  7202 - 2 ) may include two rectangular shape patches. The first rectangular shape is connected to the CPW feed line  7204 ,  7215  and may have a dimension of about 0.6 mm×3.7 mm. The second rectangular shape is capacitively coupled to an cell patch  7203 - 1 ,  7203 - 2  and may have a dimension of about 1 mm×4.8 mm. The cell patch  7203 - 1  is coupled to the launch pad  7202 - 1  through a coupling gap  7207 - 1  (e.g., 0.1524 mm) and is shorted to a ground  7210 - 2  through a via  7205 , via pad  7207  and ground line  7208 . The dimension of the cell patch  7203 - 1 , in this example, may be about 4.8 mm×7 mm. The coupling gap  7207 - 2  between the cell patch  7203 - 2  and the launch pad  7202 - 2  may have the same dimensions as the coupling gap  7207 - 1  previously mentioned. The via  7205  connects the cell patch  7203 - 1  on one top side of the substrate to a via pad  7207 , as shown in  FIG. 72D , on the bottom side of the substrate. The via  7205  connects the cell patch  7203 - 1  and via pad  7207  and may have a radius of about 0.127 mm. The center of the via pad  7207  may be located at about 3.1024 mm from the open end portion of the cell patch ( 7203 - 1 ,  7203 - 2 ). The dimension of the via pad  7207  may be about 0.8 mm×0.8 mm and is connected to the ground  7210 - 2  through an L-shape ground line  7208 . The ground line  7208  includes a first arm that is connected to the via pad  7207  and may have a dimension of about 0.3 mm×4.1 mm, and a second arm that is connected to the ground  7210 - 2  and may have a dimension of about 0.3 mm×6.35 mm.  
      The metamaterial antenna array shown in  FIGS. 72A-72E  and  73  may be measured by using a network analyzer, and the results are shown in  FIG. 74 . The results from  FIG. 74  illustrates that the metamaterial antenna array shown in  FIGS. 72A-72E  and  73  may operate at two frequencies, f1=2.57 GHz and f2=5.0 GHz to 6.0 GHz, and the coupling is about −6.0 dB and −13.0 dB at f1 and f2, respectively. Since the coupling at f2 is weak, these conditions, as mentioned example 2 in Section V, are considered in this analysis to design the vertical directional coupler.  
      A structure of the vertical directional coupler which is realized by using coupled strip lines  7513  is shown in  FIGS. 75A-75E . This vertical directional coupler may be designed on a 0.787 mm FR-4 substrate having a dielectric constant of 4.4 and four metal layers ( FIGS. 75A-75D ). In  FIG. 75E , the thicknesses of the FR-4 substrates in between layer 1   7520 - 1  and layer 2   7520 - 2 , layer 2   7520 - 2  and layer 3   7520 - 3 , and layer 3   7520 - 3  and layer 4   7520 - 4  may be 10 mil, 11 mil, and 10 mil, respectively. A coupled strip line  7513  of  FIGS. 75B and 75C  may include two overlapping strip lines printed on layer 2  ( FIG. 75B ) and layer 3  ( FIG. 75C ). In this example, the width W of the coupled strip line  7513  may be about 0.25 mm and the length L may be about 8.2 mm. The dimensions of the vertical directional coupler can be selected to have 50Ω characteristic impedance and sufficient coupling at f1 and low coupling at f2. Thus, the conditions under Eq. (21a) and Eq. (21b) are satisfied.  
      The vertical directional coupler may include four ports where P 1   7501 - 1  and P 2   7501 - 2  may be used for RF inputs, as shown in  FIGS. 75A and 75D , and ports P 3   7501 - 3  and P 4   7501 - 4  can be the outputs of the vertical directional coupler, as shown in  FIGS. 75A and 75D . Ports P 3   7501 - 3  and P 4   7501 - 4  of  FIGS. 75A and 75D  can be connected to the metamaterial antenna array shown in  FIGS. 72A-72E , as discussed in the next section. Four ends of the coupled strip line  7513  may be connected to four 1 mm×1 mm via pads ( 7510 - 2 ,  7510 - 3 ) in this example. Two CPW feed lines  7502  which are on layer 1  of  FIG. 75A  can be connected to two via pads  7510 - 2  on layer 2  of  FIG. 75B  through vias  7505 . Another pair of CPW feed lines  7503  which are on layer  4  of  FIG. 75D  may be connected to two via pads  7510 - 3  on layer 3  of  FIG. 75C  through vias  7507 .  
       FIG. 76  illustrates the simulated return loss, insertion loss, coupling, and isolation of the vertical directional coupler shown in  FIGS. 75A-75E . The results of  FIG. 76  demonstrate that the vertical directional coupler is matched well to 50Ω over a frequency range from 1 GHz to 6 GHz and has coupling of about −10 dB at 2.7 GHz and −28.5 dB coupling at 5.28 GHz.  
       FIGS. 77A-77E  shows an example in which the metamaterial antenna array illustrated in  FIGS. 72A-72E  and  FIG. 73  is connected to the outputs of the vertical directional coupler in  FIGS. 75A-75E . The CPW ( 7701 - 1 ,  7701 - 2 ,  7701 - 3 ,  7701 - 4 ) of the antenna elements in the system in  FIGS. 77A and 77D  are slightly different in shape as compared to those in the metamaterial antenna array in  FIGS. 72A-72E . This minor structural difference results from the optimization performed during the implementation. The measurement results for the dualband multi-antenna system shown in  FIG. 77  are plotted in  FIG. 78 . The results from  FIG. 78  demonstrate that the return loss better than −10 dB from about 2.4 GHz to 3.3 GHz and about 4.5 GHz to 6 GHz can be obtained while the isolations are −20.45 dB and −14 dB at 2.65 GHz and 5.58 GHz, respectively. These results further demonstrate an isolation improvement compared to the one without the coupler as shown in  FIG. 74 .  
      V.A5. Dualband Two-Element Antenna Array with 2-Way Directional Coupler Using MTM Transmission Line and LC-Network—Condition: f2≠2×f1, f2&gt;f1, Strong Coupling at f1 and Weak Coupling at f2  
      In the previous description, the dualband multi-antenna systems can be achieved by using either a conventional microwave directional coupler or a MTM coupler. The conventional microwave directional coupler used in these dualband multi-antenna system designs can either have a larger physical size which is bulky or multi-layer structure which is complicated. The MTM coupler may require multiple unit cells to satisfy the conditions in dualband operation which can have several lumped elements. In order to design a small dualband multi-antenna system which requires only a single cell MTM coupler, a LC network  7901  as shown in  FIG. 79A  can be used in the MTM coupler instead of only a single capacitor (Cm).  FIG. 79B  shows an example of using series capacitor (Cm)  7905  and series inductor (Lm)  7910  in the MTM coupler. By choosing the optimal combination of capacitor and inductor value, the frequency response of this MTM coupler can achieve high coupling at f1 and low coupling at f2.  
       FIGS. 80A-80C  shows multiple layers of a small dualband multi-antenna system which may include two metamaterial antennas and a MTM coupler. The small dualband multi-antenna system shown in  FIGS. 80A-80C  may be constructed on a 1 mm FR-4 substrate  8060  with dielectric constant of 4.4. As illustrated in  FIG. 80A  and  FIG. 80B , each metamaterial antenna may include a top patch  8001 , launch pad  8005 , via  8010 , via pad  8015  and a via line  8020 . The antenna is excited by a 50Ω antenna feed  8040  which is printed on layer 1   8030  and layer 2   8035  and connected by a metallic via  8010 . One side of the launch pad  8005  is connected to the antenna feed  8040  and the other side is coupled to the top patch  8001  through a coupling gap  8007 . The top patch  8001  is connected to the via pad  8015  on the other side of the substrate by using a metallic via  8010 . The via pad  8015  is connected to the CPW ground  8050 - 1  through the via line  8020 . The four ports MTM coupler can include two metamaterial transmission lines and a LC network connecting in between. Each metamaterial transmission line may include a CPW feed  8025 , series capacitor (CL)  8055 , and a CPW shorted stub  8060 . One end portion of the series capacitor (CL)  8055  is connected to the antenna feed  8040  and the other end portion is connected to the CPW feed  8025  and CPW shorted stub  8060 . One end portion of the CPW shorted stub  8060  may be connected to the CPW ground  8050 - 1  and the other end portion may be connected to the CPW feed  8025 . The LC network, in this implementation, may include a series capacitor (Cm)  8065  and a series inductor (Lm)  8070 . One end portion of the Cm  8065  may be connected to CPW feed  8025  while the other end portion can be connected to Lm  8070 . Similarly, one end portion of the Lm  8070  can be connected to another CPW feed  8025  while the other end portion can be connected to Cm  8065 . Values for Cm and Lm may be selected to be about 0.4 pF and 6.8 nH, respectively.  
       FIG. 81  illustrates the simulated return losses and coupling of the small dualband multi-antenna system shown in  FIGS. 80A-80C . The results of  FIG. 81  demonstrate that the isolation is better than about −10 dB in the low band (2.77 GHz to 2.9 GHz) and high band (4.72 GHz to 6.0 GHz) while still maintaining sufficient impedance matching at both bands.  
      VI. Multi-Antenna, Directional Coupler System: 2-Way Forward Wave MTM Coupler  
      An MTM coupler can be modeled using the general equivalent circuit depicted in  FIG. 69 , where L m  and C m  are the induced mutual coupling by the microstrip coupled lines, CPW coupled lines or other type of coupled transmission lines in the planar form or in the 3-D form. These parameters have already been introduced for the MTM coupler represented by the equivalent circuit in  FIG. 69 . To extend the analysis for a general case, we use additional capacitive coupling by inserting a capacitor C m1  between the two coupled lines, and additional inductive coupling by inserting an inductor L m1  between the two coupled lines as shown in  FIG. 82A . These additional coupling components can be used to manipulate the MTM coupler between backward-wave (BW) and forward-wave (FW) coupling as well as to create high coupling in some bands and low coupling in other bands. Like other components, L m1  and C m1  can be implemented as discrete components or distributed structures.  
      The following analysis provides a way to estimate a range of C m1  and L m1  values as well as C L  and L L  required for achieving necessary couplings at specific bands given a specific type, length, and impedance of coupled transmission lines. It may be still necessary to simulate the whole structure for final tuning and optimization. The analysis described in “Generalized Coupled-Mode Approach of Metamaterial Coupled-Line Couplers: Coupling Theory, Phenomenological Explanation, and Experimental Demonstration”, IEEE Transactions on Microwave Theory and Techniques, Vol. 55, No. 5, May 2007 can be followed along with making a modification of including additional C m1  and L m1  to C m  and L m , which are the mutual coupling parameters due the coupled lines. In this analysis, only the symmetric line case is considered.  
      The theoretical BW and FW coupling factors KBW and KFW are given by:  
             KFW   =       1   2     ⁢   ω   ⁢         L   R     ·     C   R         ⁢     (           L   m     +     L     m   ⁢           ⁢   1           L   R       -         C   m     +     C     m   ⁢           ⁢   1           C   R         )               Eq   .           ⁢     (     23   ⁢           ⁢   a     )                 KBW   =       1   2     ⁢   ω   ⁢         L   R     ·     C   R         ⁢     (           L   m     +     L     m   ⁢           ⁢   1           L   R       +         C   m     +     C     m   ⁢           ⁢   1           C   R         )               Eq   .           ⁢     (     23   ⁢           ⁢   b     )               
 
      The FW a +   1  (and a +   2 ) and BW a −   1  (and a −   2 ) waves along the 1 st  (and 2 nd ) metamaterial transmission lines ( 8221 - 1 ,  8221 - 2 ) shown in  FIG. 82B  are given by the formula below, where z is the position along the metamaterial transmission lines ( 8221 - 1 ,  8221 - 2 ):  
                     ⁢         a   1   +     ⁡     (   z   )       =       A   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢     β   I     ⁢   z         +     B   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢     β   II     ⁢   z         +     C   ⁢           ⁢     ⅇ       +   j     ⁢           ⁢     β   I     ⁢   z         +     D   ⁢           ⁢     ⅇ       +   j     ⁢           ⁢     β   II     ⁢   z                     Eq   .           ⁢     (     24   ⁢           ⁢   a     )                         ⁢         a   2   +     ⁡     (   z   )       =       A   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢     β   I     ⁢   z         -     B   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢     β   II     ⁢   z         +     C   ⁢           ⁢     ⅇ       +   j     ⁢           ⁢     β   I     ⁢   z         -     D   ⁢           ⁢     ⅇ       +   j     ⁢           ⁢     β   II     ⁢   z                     Eq   .           ⁢     (     24   ⁢           ⁢   b     )                     a   1   -     ⁡     (   z   )       =         A   ⁡     (       β   -     β   I     +   KFW     KBW     )       ⁢     ⅇ       -   j     ⁢           ⁢     β   I     ⁢   z         +       B   ⁡     (       β   -     β   II     -   KFW     KBW     )       ⁢     ⅇ       -   j     ⁢           ⁢     β   II     ⁢   z         +       C   ⁡     (       β   +     β   I     +   KFW     KBW     )       ⁢     ⅇ       +   j     ⁢           ⁢     β   I     ⁢   z         +       D   ⁡     (       β   +     β   II     -   KFW     KBW     )       ⁢     ⅇ       +   j     ⁢           ⁢     β   II     ⁢   z                   Eq   .           ⁢     (     24   ⁢           ⁢   c     )                     a   2   -     ⁡     (   z   )       =         A   ⁡     (       β   -     β   I     +   KFW     KBW     )       ⁢     ⅇ       -   j     ⁢           ⁢     β   I     ⁢   z         -       B   ⁡     (       β   -     β   II     -   KFW     KBW     )       ⁢     ⅇ       -   j     ⁢           ⁢     β   II     ⁢   z         +       C   ⁡     (       β   +     β   I     +   KFW     KBW     )       ⁢     ⅇ       +   j     ⁢           ⁢     β   I     ⁢   z         -       D   ⁡     (       β   +     β   II     -   KFW     KBW     )       ⁢     ⅇ       +   j     ⁢           ⁢     β   II     ⁢   z                   Eq   .           ⁢     (     24   ⁢           ⁢   d     )               
 
 where, β is the propagation constant of a single uncoupled metamaterial transmission line, β I  &amp; β II  are the propagation constants of the coupled metamaterial transmission lines for even and odd modes, and are all given by the following relationships:  
               β   =     ω   ⁢         L   R     ·     C   R         ⁢     (     1   -     ω     ω   0         )     ⁢           ⁢   where       ⁢     
     ⁢       ω   0     =       1         L   R     ·     C   R           =     1         L   R     ·     C   L                       Eq   .           ⁢     (     25   ⁢           ⁢   a     )               
 
 and uncoupled metamaterial transmission line impedances  
             Z   =           L   L           C   L         =         L   R           C   R                                     β   I     =           (     KFW   +   β     )     2     -     KBW   2                 Eq   .           ⁢     (     25   ⁢           ⁢   b     )                   β   II     =           (     KFW   -   β     )     2     -     KBW   2                 Eq   .           ⁢     (     25   ⁢           ⁢   c     )               
 
      The scattering parameters of the MTM coupler are defined as follows:  
               S   ⁢           ⁢   11     =         a   1   -     ⁡     (     z   =   0     )           a   1   +     ⁡     (     z   =   0     )                 Eq   .           ⁢     (     26   ⁢           ⁢   a     )                   S   ⁢           ⁢   12     =       S   ⁢           ⁢   34     =         a   1   +     ⁡     (     z   =   L     )           a   1   +     ⁡     (     z   =   0     )                   Eq   .           ⁢     (     26   ⁢           ⁢   b     )                   S   ⁢           ⁢   13     =       S   ⁢           ⁢   24     =         a   2   -     ⁡     (     z   =   0     )           a   1   +     ⁡     (     z   =   0     )                   Eq   .           ⁢     (     26   ⁢           ⁢   c     )                   S   ⁢           ⁢   14     =       S   ⁢           ⁢   23     =         a   2   +     ⁡     (     z   =   L     )           a   1   +     ⁡     (     z   =   0     )                   Eq   .           ⁢     (     26   ⁢           ⁢   d     )               
 
 where, L is the total length of one MTM coupler unit cell as shown in  FIG. 82A-82B . 
 
      The boundary conditions that determine the constant A, B, C, and D in Eq. (24a-24d) are as follows: 
 
 a   1   + ( z= 0)= a   0   Eq. (27a) 
 
 a   2   + ( z= 0)= a   1   − ( z=L )= a   2   − ( z=L )=0  Eq. (27b) 
 
 Using the above equations, the parameter values such as L R , C R , etc. for an MTM coupler with given coupled lines can be obtained. Thereafter, the scattering matrix S ij  that defines the coupling levels and coupler operating bands can be obtained. 
 
      The approach presented in this section is for the case where coupling occurs in the forward direction instead of backward direction as in the examples previously presented. In general, symmetric-line couplers as shown in  FIG. 82A  can couple signals between port 1   8201 - 1  and port 4   8201 - 4  when S 141  is high and |S 13 | is low in Eq. (26a-26d), where |S 14 | is given by:  
               S   ⁢           ⁢   14     =       -   j     ⁢           ⁢     ⅇ     (       -   j     ⁢         β   I     -     β   II       2     ⁢   L     )       ⁢     sin   ⁡     (           β   I     -     β   II       2     ⁢   L     )                 Eq   .           ⁢     (     28   ⁢           ⁢   a     )                          S   ⁢           ⁢   14          2     =       sin   2     ⁡     (           β   I     -     β   II       2     ⁢   L     )               Eq   .           ⁢     (     28   ⁢           ⁢   b     )               
 
      Most of the TEM transmission line type symmetric couplers have KBW&gt;&gt;KFW in Eq. (23a-23b) because L m /L R  is close to C m /C R  in value. Thus, the relationship β I ≈β II  from Eq. (25a-25c) leads |S 14 | to near zero. Therefore, most, if not all, conventional directional couplers are generally BW in nature. In MTM coupler, the propagation constants β I  and β II  can be different depending on the values of L m1  and C m1  for a given coupled line designed with C R , L R , C m , and L m . Therefore, the following free parameters C L , C m1 , and/or L m1  may be used to tune and optimize the length L  8205  and coupling level at specific frequency f. Notably, in this case, FW coupling can occur in a MTM coupler when (L m1 +L m )/L R &gt;&gt;(C m1 +C m )/C R . One example of planar MTM coupler with FW coupling will be demonstrated in  FIG. 82C  in the following description.  
      The asymmetric MTM coupler can be also implemented by paralleling two metamaterial transmission lines ( 8241 - 1 ,  8241 - 2 ) as shown  FIG. 82D . In this analysis, C L1 , C L2 , L L1 , and L L2  are used to differentiate LH portion of the two parallel metamaterial transmission lines ( 8241 - 1 ,  8241 - 2 ) where 1 indicates the 1 st  metamaterial transmission line ( 8241 - 1 ) and 2 indicates the 2 nd  metamaterial transmission line ( 8241 - 2 ). The following analysis can provide a way to estimate a range of C m1  and L m1  values as well as required C L1 , C L2 , L L1 , and L L2  to achieve necessary couplings at specific bands. It may be still necessary to simulate the final structure for final tuning and optimization. The analysis described in “Generalized Coupled-Mode Approach of Metamaterial Coupled-Line Couplers: Coupling Theory, Phenomenological Explanation, and Experimental Demonstration”, IEEE Transactions on Microwave Theory and Techniques, Vol. 55, No. 5, May 2007 can be followed along with making a modification of including the additional C m1  and L m1  to C m  and L m , which are the mutual coupling parameters due to the coupled lines. The theoretical BW and FW coupling factors KBW and KFW are given by:  
             KFW   =       1   2     ⁢       ω   ⁡     (       L     R   ⁢           ⁢   1       ⁢     L     R   ⁢           ⁢   2       ⁢     C     R   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   2         )       14     ⁢     (                 L   m     +     L     m   ⁢           ⁢   1               L     R   ⁢           ⁢   1       ⁢     L     R   ⁢           ⁢   2             -                   C   m     +     C     m   ⁢           ⁢   1               C     R   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   2                   )               Eq   .           ⁢     (     30   ⁢           ⁢   a     )                 KBW   =       1   2     ⁢   ω   ⁢       (       L     R   ⁢           ⁢   1       ⁢     L     R   ⁢           ⁢   2       ⁢     C     R   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   2         )     14     ⁢     (                 L   m     +     L     m   ⁢           ⁢   1               L     R   ⁢           ⁢   1       ⁢     L     R   ⁢           ⁢   2             +                   C   m     +     C     m   ⁢           ⁢   1               C     R   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   2                   )               Eq   .           ⁢     (     30   ⁢           ⁢   b     )               
 
      The FW a +   1 , (and a +   2 ) and a BW a −   1 , (and a −   2 ) waves along the 1 st  (and 2 nd ) metamaterial transmission line are given by the formula below, where z is the position along the MTM coupler: 
 
 a   1   + ( z )= Ae   −jβ     I     z   +Be   −jβ     II     z   +Ce   +jβ     I     z   +De   +jβ     II     z   Eq. (31a) 
 
 a   2   + ( z )= A   2   e   −jβ     I     z   +B   2   e   −jβ     II     z   +C   2   e   +jβ     I     z   +D   2   e   +jβ     II     z   Eq. (31b) 
 
 a   1   − ( z )= A   1   ′e   −jβ     I     z   +B   1   ′e   −jβ     II     z   +C   1   ′e   +jβ     I     z   +D   1   ′e   +jβ     II     z   Eq. (31c) 
 
 a   2   − ( z )= A   2   ′e   −jβ     I     z   +B   2   ′e   −jβ     II     z   +C   2   ′e   +jβ     I     z   +D   2   ′e   +jβ     II     z   Eq. (31d) 
 
      Here, the coefficients can be expressed in terms of A, B, C, and D as:  
               A   3   ′     =           A   1     ⁡     (       β   1     -     β   I       )       +     KFW   ⁢           ⁢     A   3         KBW             Eq   .           ⁢     (     32   ⁢           ⁢   a     )                   B   3   ′     =           B   1     ⁡     (       β   1     -     β   II       )       +     KFW   ⁢           ⁢     B   3         KBW             Eq   .           ⁢     (     32   ⁢           ⁢   b     )                   C   3   ′     =           C   1     ⁡     (       β   1     +     β   I       )       +     KFW   ⁢           ⁢     C   3         KBW             Eq   .           ⁢     (     32   ⁢           ⁢   c     )                   D   3   ′     =           D   1     ⁡     (       β   1     +     β   II       )       +     KFW   ⁢           ⁢     D   3         KBW             Eq   .           ⁢     (     32   ⁢           ⁢   d     )                   A   1   ′     =           A   3     ⁢     (       β   3     -     β   I       )       +     KFW   ⁢           ⁢     A   1         KBW             Eq   .           ⁢     (     33   ⁢           ⁢   a     )                   B   1   ′     =           B   3     ⁡     (       β   3     -     β   II       )       +     KFW   ⁢           ⁢     B   1         KBW             Eq   .           ⁢     (     33   ⁢           ⁢   b     )                   C   1   ′     =           C   3     ⁡     (       β   3     +     β   I       )       +     KFW   ⁢           ⁢     C   1         KBW             Eq   .           ⁢     (     33   ⁢           ⁢   c     )                   D   1   ′     =           D   3     ⁡     (       β   3     +     β   II       )       +     KFW   ⁢           ⁢     D   1         KBW             Eq   .           ⁢     (     33   ⁢           ⁢   d     )                   A   2     =     A   ⁢       2   ⁢           ⁢   KFW   ⁢           ⁢     β   1             -     (       β   1     +     β   I       )       ⁢     (       β   2     -     β   I       )       +     KFW   2     -     KBW   2                   Eq   .           ⁢     (     34   ⁢           ⁢   a     )                   B   2     =     B   ⁢       2   ⁢           ⁢   KFW   ⁢           ⁢     β   1             -     (       β   1     +     β   II       )       ⁢     (       β   2     -     β   II       )       +     KFW   2     -     KBW   2                   Eq   .           ⁢     (     34   ⁢           ⁢   b     )                   C   2     =     C   ⁢       2   ⁢           ⁢   KFW   ⁢           ⁢     β   1             -     (       β   1     +     β   I       )       ⁢     (       β   2     -     β   I       )       +     KBW   2     -     KFW   2                   Eq   .           ⁢     (     34   ⁢           ⁢   c     )                   D   2     =     D   ⁢       2   ⁢           ⁢   KFW   ⁢           ⁢     β   1             -     (       β   1     +     β   II       )       ⁢     (       β   2     -     β   II       )       +     KFW   2     -     KBW   2                   Eq   .           ⁢     (     34   ⁢           ⁢   d     )               
 
 where, β 1  and β 2  are the propagation constants of the two uncoupled metamaterial transmission lines ( 8241 - 1 ,  8241 - 2 ) and β I /β II  are the propagation constants of the metamaterial coupled lines even and odd modes and are all given as follows:  
                 β   1     =     ω   ⁢         L     R   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   1           ⁢     (     1   -       ω     β   ⁢           ⁢   1       ω       )     ⁢           ⁢   where       ⁢     
     ⁢       ω     β   ⁢           ⁢   1       =       1         L     L   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   1             =       1         L             ⁢     R   ⁢           ⁢   1         ⁢     C     L   ⁢           ⁢   1             ⁢           ⁢   and         ⁢     
     ⁢       uncoupled   ⁢           ⁢   line   ⁢           ⁢   impedance   ⁢           ⁢   Z     =           L     L   ⁢           ⁢   1         C     L   ⁢           ⁢   1           =         L     R   ⁢           ⁢   1         C     R   ⁢           ⁢   1                       Eq   .           ⁢     (     35   ⁢           ⁢   a     )                     β   2     =     ω   ⁢         L     R   ⁢           ⁢   1       ⁢     C     R   ⁢           ⁢   1           ⁢     (     1   -       ω     β   ⁢           ⁢   2       ω       )     ⁢           ⁢   where       ⁢     
     ⁢       ω     β   ⁢           ⁢   2       =       1         L     L   ⁢           ⁢   2       ⁢     C     R   ⁢           ⁢   2             =       1         L             ⁢     R   ⁢           ⁢   2         ⁢     C     L   ⁢           ⁢   2             ⁢           ⁢   and         ⁢     
     ⁢       uncoupled   ⁢           ⁢   line   ⁢           ⁢   impedance   ⁢           ⁢   Z     =           L     L   ⁢           ⁢   2         C     L   ⁢           ⁢   2           =         L     R   ⁢           ⁢   2         C     R   ⁢           ⁢   2                       Eq   .           ⁢     (     35   ⁢           ⁢   b     )                   β   I     =               KFW   2     -     KBW   2     +         β   1   2     +     β   3   2       2     +                           (           ⁢               ⁢       β             ⁢   1               ⁢   2       ⁢           -           ⁢     β             ⁢   3               ⁢   2                     ⁢   2       )     2     -           ⁢         KBW             ⁢   2       ⁡     (       β             ⁢   1       -     β             ⁢   3         )       2     +                       ⁢         KFW             ⁢   2       ⁡     (       β             ⁢   1       ⁢           +           ⁢     β             ⁢   3         )       2                               Eq   .           ⁢     (     35   ⁢           ⁢   c     )                   β   II     =               KFW   2     -     KBW   2     +         β   1   2     +     β   3   2       2     -                           (           ⁢               ⁢       β             ⁢   1               ⁢   2       ⁢           -           ⁢     β             ⁢   3               ⁢   2                     ⁢   2       )     2     -           ⁢         KBW             ⁢   2       ⁡     (       β             ⁢   1       -     β             ⁢   3         )       2     +                       ⁢         KFW             ⁢   2       ⁡     (       β             ⁢   1       +     β             ⁢   3         )       2                               Eq   .           ⁢     (     35   ⁢           ⁢   d     )               
 
 Thus, the scattering parameters of the directional couplers are defined by:  
               S   ⁢           ⁢   11     =         a   1   -     ⁡     (     z   =   0     )           a   1   +     ⁡     (     z   =   0     )                 Eq   .           ⁢     (     36   ⁢           ⁢   a     )                   S   ⁢           ⁢   12     =         a   1   +     ⁡     (     z   =   L     )           a   1   +     ⁡     (     z   =   0     )                 Eq   .           ⁢     (     36   ⁢           ⁢   b     )                   S   ⁢           ⁢   13     =         a   3   -     ⁡     (     z   =   0     )           a   1   +     ⁡     (     z   =   0     )                 Eq   .           ⁢     (     36   ⁢           ⁢   c     )                   S   ⁢           ⁢   14     =         a   3   +     ⁡     (     z   =   L     )           a   1   +     ⁡     (     z   =   0     )                 Eq   .           ⁢     (     36   ⁢           ⁢   d     )               
 
 The boundary conditions that determine the constant A, B, C, and D in Eq. (31a-31d) are: 
 
 a   1   + ( z= 0)= a   0   a   3   + ( z= 0)= a   1   − ( z=L )= a   3   − ( z=L )=0  Eq. (37) 
 
 Where L is the total length of one MTM coupler unit cell. For a given coupled lines determined by L R1 , C R1 , L R2 , C R2 , L m , and C m  and using Eq. (30a-30b) to Eq. (36a-36d); the scattering matrix Sij that can determine coupling levels and coupler operating bands may be manipulated using the free parameters C L1  (or L L1 ), C L2  (or L L2 ) and C m1  and/or L m1 . 
 
      In this section, two examples of FW MTM couplers are considered. One example is a planar FW MTM directional coupler. The schematic of this coupler is shown in  FIG. 82C . The planar FW MTM directional coupler  8200   c  shown in  FIG. 82C  can be implemented by paralleling two metamaterial transmission lines ( 8247 - 1 ,  8247 - 2 ) with an additional inductor L m1  (Cm 1  is 0 in this example) connecting between the two metamaterial transmission lines ( 8247 - 1 ,  8247 - 2 ). Each metamaterial transmission line ( 8247 - 1 ,  8247 - 2 ) has two unit cells ( 8233 - 1 ,  8233 - 2 ). Each metamaterial unit cell ( 8233 - 1  and  8233 - 2 ) comprises two transmission lines (represented by a gray rectangular boxes  8238  in  FIG. 82C ), two series capacitors of 2C L  and one shunt inductor of L L . This FW MTM coupler can be fabricated on a FR-4 substrate having a dielectric constant of about 4.4 and thickness of about 0.787 mm. Each of the transmission line  8238  can have an intrinsic series inductance L R  and a shunt capacitance C R . Therefore, the implemented planar FW directional coupler in  FIG. 82C  can be represented by the equivalent circuit of  FIG. 82A . The mutual inductor capacitor C m  shown in  FIG. 82A  is induced when the two metamaterial transmission lines ( 8247 - 1 ,  8247 - 2 ) are within close proximity.  
      Another example of FW MTM coupler is a vertical FW MTM coupler shown in  FIGS. 83A-83D . This FW MTM coupler may be realized by cascading two coupled metamaterial unit cells. In  FIG. 83A-83D , each coupled metamaterial cell is built by paralleling two metamaterial unit cells vertically with an additional inductor L m1  connecting between the two metamaterial unit cells, wherein one set of unit cells is on the top layer  8325  of the substrate (between top layer  8325  and bottom layer  8330 ), the other set of unit cells is on the bottom layer  8330  of the substrate (between top layer  8325  and bottom layer  8330 ), and the inductors L m1    8340  couple the top and bottom layers as shown in  FIG. 83B . Each metamaterial unit cell also comprises two transmission lines  8303 - 1 , two series capacitors 2C L    8310  and one shunt inductor L L    8305 . The vertically coupled transmission lines (paralleling transmission line  8303 - 1  and  8303 - 2 ) provide mutual inductance L m  and mutual capacitance C m . In addition, each port (P 1   8301 - 1 , P 2   8301 - 2 , P 3   8301 - 3 , P 4   8301 - 4 ) of the vertical FW MTM coupler is connected to the transmission lines  8303 - 1 ,  8303 - 2  through a CPW line ( 8320 - 1 ,  8320 - 2 ,  8320 - 3 ,  8320 - 4 ).  
      The planar FW MTM coupler shown in  FIG. 82C  is designed to have FW coupling at 2.4 GHz.  
      Some of the design parameters for the planar FW MTM coupler shown in  FIG. 82C  are summarized in Table 10:  
               TABLE 10                       Planar FW MTM Coupler                                                    w   1.5   mm           s   0.1   mm           L   8   mm           C m     0.2444   pF           C R     0.936   pF           L R     2.18   nH           L m     0.5416   nH                      
 
      The planar FW MTM coupler is simulated by using Ansoft Designer. In  FIGS. 84A-84C , the simulation results for the planar FW MTM coupler are presented. For a fixed L m1 =7nH and length L=8 mm, C L  can be varied to change the coupling level at 2.4 GHz. In  FIGS. 85A-85D  the value of C L =5.6 pF is fix and the value of L m1  is varied. The coupling level at 2.4 GHz can be changed according to  FIGS. 85A-85D .  
      Another example of the vertical FW MTM coupler shown in  FIGS. 83A-83D  is simulated by using Ansoft HFSS where the simulated results are shown in  FIG. 86 . The frequency and FW coupling at lower frequency band of the vertical FW coupler can be found to be almost the same as those of the planar FW coupler shown in  FIGS. 82A-82D . However, the FW coupling at higher band of the vertical FW coupler is found to be significantly different from that of the planar FW coupler. Furthermore, the coupling levels and bands can be found to be nearly the same between the case of using the planar or vertical coupled microstrip lines and the case of using the coupled CPW.  
      VI.A. Dualband Two-Element Antenna Array with 2-Way Vertical Forward Wave MTM Coupler—Condition: f2≠2×f1, f2&gt;f1, Strong Coupling at f1 and Weak Coupling at f2  
       FIGS. 87A-87B  depicts another example of dualband multi-antenna system, which integrates a metamaterial antenna array  8700 - 1  and a vertical FW MTM coupler  8700 - 2 . One of the antennas in the array is printed on top of the substrate  8710  and the other one is printed on bottom of the substrate  8710 . In  FIG. 87A , the inputs for the antenna array, port 1 ′  8705 - 1  and port 2 ′  8705 - 2 , can be connected to port 3   8701 - 3  and port 2   8701 - 2  of the vertical FW MTM coupler  8700 - 2 , respectively. This antenna array can exhibit high coupling at about 2.4 GHz band and low coupling at about 5 GHz band. The same phase analysis may be followed as in example 2 in Section V and find that the phase constraints are as follows: 
 θ2=Phase( S 12)=θ4=Phase( S 34)  Eq. (29a)  θ3=Phase(Antenna  S 1′2′)  Eq. (29b)  θ4=Phase( S 43)  Eq. (29c)  θ2=Phase( S 14)  Eq. (29d)  θ2+θ3+θ4−θ1=180°  Eq. (29e)  2θ2−θ1=−180°−θ3  Eq. (29f)  
      Additional details of the vertical FW MTM coupler  8700 - 2 , as shown in  FIG. 87A , are illustrated in  FIGS. 88A-88C  and  89 A- 89 D. The transmission paths are from p 1   8801 - 1  to P 2   8801 - 2  and from p 3   8801 - 3  to P 4   8801 - 4 . The FW coupling paths are from P 1   8801 - 1  to P 4   8801 - 4  and from P 2   8801 - 2  to P 3   8801 - 3 . The vertical FW MTM coupler can be implemented on a multi-layer FR4 substrate comprising three dielectric layers and four metal layers, as shown in  FIG. 88B . Each dielectric layer measures the height of 10 mil. Based on the analysis on the planar and vertical FW MTM couplers described in the previous section, the parameter values for this vertical coupler may be obtained to be nearly the same as in the previous examples with the exception of C L =2 pF, L L =18 nH and L m1 =7.5 nH.  
       FIG. 90  shows the simulation results of the vertical FW MTM coupler used in the dualband multi-antenna system shown in  FIGS. 87A-87B . As noted earlier, the FW coupling is high at 2.4 GHz and low at 5 GHz. There is no BW coupling which is between P 1   8801 - 1  and P 3   8801 - 3  or between P 2   8801 - 2  and P 4   8801 - 4  (isolation shown in  FIG. 90 ) at both 2.4 GHz and 5 GHz.  
       FIGS. 91A-91C  shows the structure of the dualband metamaterial antenna array used in the dualband multi-antenna system shown in  FIG. 87A-87B . Two antenna elements are on different sides of the substrate.  
       FIG. 92  shows the simulation results of the metamaterial antenna array shown in  FIG. 91 . It can be seen from  FIG. 92  that the coupling is high at about 2.4 GHz (near −6 dB) and low at about 5 GHz.  
       FIG. 93  shows the simulation results of the dualband multi-antenna system shown in  FIG. 87 . The results of  FIG. 93  demonstrate that the coupler can improve the coupling at about 2.5 GHz to −15 dB without affecting the 5 GHz band. The bandwidth coverage may still be adequate at about 2.5 GHz.  
      VII. Multi-Antenna, Directional Coupler System: WiFi and WiMax Antenna Array, 2-Way Directional Coupler  
      A directional coupler may be used to improve the isolation across a WiFi and WiMax frequency bands. By reducing the isolation between the WiFi and WiMax antennas, the interference between the WiFi and WiMax signals can be minimized. A multi-band multi-antenna system shown  FIG. 94  may include a multi-band metamaterial antenna array ( 9425 ,  9430 ) and a directional coupler  9415 . The multi-band metamaterial antenna array may include a metamaterial WiFi antenna  9430  and a metamaterial WiMax antenna  9425 . The WiFi antenna  9430  may include a port P 2 ′  9415 - 2  and can have a frequency range that varies from about 2.4 GHz to 2.48 GHz. The WiMax antenna  9425  may include a port P 1 ′  9415 - 1  and can have a frequency range that varies from about 2.5 GHz to 2.7 GHz. As shown in  FIG. 94 , the spacing, d  9420 , between the WiFi and WiMax antennas can be used to determine the magnitude and phase of the coupling between the two antenna elements ( 9425 ,  9430 ).  
      The directional coupler  9415  shown in  FIG. 94  can be a four port passive device. In one implementation, the directional coupler may include input ports P 1   9410 - 1  and P 3   9410 - 3  and output ports P 2   9410 - 2  and P 4   9410 - 4 . Each input port may be assigned to a specific signal and each output port may be assigned to a specific antenna that is coupled to the directional coupler  9415 . For example, P 1   9410 - 1  can be the input port of a WiMax signal  9401 , P 3   9410 - 3  can be the input port of a WiFi signal  9405 , P 2   9410 - 2  can be the output port of the directional coupler  9415  connected to the WiMax antenna  9425 , and P 4   9410 - 4  can be the output port of the directional coupler  9415  connected to the WiFi antenna  9430 .  
      As shown in  FIG. 94 , the WiMax signal  9401  can be coupled from the input port P 1   9410 - 1  to the input port P 3   9410 - 3  through two paths. The first path can be traced from the input port P 1   9410 - 1  to the input port P 3   9410 - 3  via the coupling of the directional coupler  9415 . The second path can be traced starting at the input port P 1   9410 - 1 . From the input port P 1   9410 - 1 , the second path can be traced to the output port P 2   9410 - 2  via the transmission of the directional coupler  9415 . From the output port P 2   9410 - 2 , the second path can be further traced to the WiMax antenna port P 1 ′  9415 - 1 . From the WiMax antenna port P 1 ′  9415 - 1 , the second path can be traced to the WiFi antenna port P 2 ′  9415 - 2  via the coupling between the WiMax  9425  and WiFi  9430  antennas. From the WiFi antenna port P 2 ′  9415 - 2 , the second path can be traced to the output port P 4   9410 - 4 . From the output port P 4   9410 - 4 , the second path can be traced to the input port P 3   9410 - 3  via the transmission of the directional coupler  9415 . When the signals from the two paths merge at the input port P 3   9410 - 3  and have the same magnitude and 180° phase difference, the isolation between the WiFi  9425  and WiMax  9430  antennas can be maximized. Therefore, maximizing the isolation between the WiFi and WiMax antennas can be achieved by properly designing the directional coupler and antennas. For directional couplers, several approaches are generally available for achieving optimum isolation requirements. In next section, a microwave coupled line coupler and metamaterial directional coupler for improving isolation and system performance are presented.  
      VII.A Multi-Antenna, Directional Coupler System: WiFi and WiMax Antenna Array  
      In yet another implementation of a multi-band multi-antenna system, an exemplary multi-band metamaterial antenna array supporting frequency bands used in WiMax and WiFi systems is illustrated in  FIGS. 95A-95F  and  FIG. 96 . The multi-band antenna array can be designed on a FR-4 substrate. The four-layer FR-4 substrate can include three substrate layers in which each substrate layer has a dielectric constant of 4.4. As shown in  FIG. 96 , the three substrate layers are denoted as substrate I  9630 , substrate II  9635 , and substrate III  9640 , and may be 0.254 mm, 1.0668 mm, and 0.254 mm in thickness, respectively. Substrates I, II, and III are also depicted in  FIGS. 95A-95F . For example, substrate I include elements  9521  and  9536  as illustrated in  FIGS. 95A and 95B , respectively. Substrate II include elements  9546  and  9556  as illustrated in  FIGS. 95C and 95D , respectively. Substrate III include elements  9566  and  9576  as illustrated in  FIGS. 95E and 95F , respectively. Each substrate may have a width and length that measures 80 mm and 49 mm, respectively. Illustrations of the top and bottom views of each substrate are shown in  FIGS. 95A-95F . In addition to the three substrates, the multi-band metamaterial antenna array shown in  FIG. 95A  may include two antenna elements, a metamaterial WiMax antenna  9501  and a metamaterial WiFi antenna  9503 , which can be located at the edge of the substrate  19521 . The spacing, d  9524 , between the two antennas may be 45 mm as shown in  FIG. 95A .  
      As shown in  FIG. 96 , the metamaterial WiMax antenna  9605  may include a cell patch  9601 , a launch pad  9610 , a via  9615 , a via pad  9625 , and a via line  9620 . Referring to  FIG. 95A , the cell patch  9506  of the WiMax antenna  9501  can be formed on the top side portion of substrate  19521 . In  FIG. 96 , the via pad  9625  can be formed on the bottom side portion of substrate III  9640 . The cell patch  9601  can be connected to the via pad  9625  through a metallic via  9615  and can have a dimension of about 3.2 mm×6.2 mm as shown in  FIG. 96 . In reference to the via location, the via may be positioned about 3.575 mm away from the top edge portion of the cell patch  9506  and 1.6 mm away from the side edge portion of the cell patch  9506  as illustrated in  FIG. 95A . In reference to the via and the via pad physical dimensions, the via radius may be about 0.125 mm, and the via pad dimension may be about 0.762 mm×1 mm. In  FIG. 96 , the via pad  9625  may be connected to a coplanar waveguide (CPW) ground, CPW ground IV  9660 , through the via line  9620 . The via line  9620  can be attached at the center of the via pad  9625  and may have a dimension of about 6.7 mm×0.2032 mm. Referring to the cell patch  9506  and the launch pad  9512  of the WiMax antenna  9501  of  FIG. 95A , the cell patch  9506  can be coupled to the launch pad  9512  through a coupling gap  9507  that measures about 0.1 mm in width. The launch pad  9512  of the WiMax antenna  9501  may include two rectangular patches. The first rectangular patch may be about 1.5 mm in length and have the same width as the cell patch  9506 , and the second rectangular patch may have a dimension of about 0.3 mm×3 mm. As shown in  FIG. 95A , the first rectangular patch can be coupled to the cell patch  9506  of the WiMax antenna  9501  while the second rectangular patch can be coupled to a 50Ω CPW feed line  9515 . The dimension of the 50Ω CPW feed line  9515  connected to the WiMax antenna  9501  may be about 0.4 mm×5 mm with a gap of 0.2 mm to the CPW ground  19518 .  
      As illustrated in  FIGS. 95A-95F  and  FIG. 96 , the metamaterial WiFi antenna  9501  of the multi-band antenna array may include a cell patch  9506 , a launch pad  9512 , a via  9509 , a via pad  9625  and a via line  9620 . Referring again to  FIG. 96 , the cell patch  9601  of the WiFi antenna  9603  can be formed on the top side portion of substrate  19630 , and the via pad  9625  can be formed on the bottom side portion of substrate III  9640 . The cell patch  9601  can be connected to the via pad  9625  through a metallic via  9615  and may have a dimension of about 3.2 mm×7.3 mm. In reference to the via location, the via  9615  may be positioned about 3.175 mm away from the top edge portion of the cell patch  9601  of WiFi antenna  9603  and about 1.6 mm away from the side edge portion of the cell patch  9601  of WiFi antenna  9603 . In reference to the physical dimensions of the via  9615  and the via pad  9625 , the via radius may be about 0.125 mm, and the via pad  9625  can be about 0.762 mm×1 mm. The via pad  9625  can be connected to a CPW ground, CPW ground IV  9660 , through the via line  9620  as shown in  FIG. 96 . The via line  9620  can be attached at the center of the via pad  9625  and may have a dimension of about 8.1 mm×0.2032 mm. Referring the WiFi antenna  9503  of  FIG. 95A , the cell patch  9506  can be coupled to the launch pad  9512  through a coupling gap which may be about 0.1 mm. The launch pad  9512  of the WiFi antenna  9503  may include two rectangular patches. The first rectangular patch may be 1.5 mm in length and have the same width as the cell patch  9506 , and the second rectangular patch may have a dimension of about 0.3 mm×3 mm. As shown in  FIG. 95A , the first rectangular patch can be coupled to the cell patch  9506  of the WiFi antenna  9503  while the second rectangular patch can be coupled to a 50Ω CPW feed line  9515 . The dimension of the 50Ω CPW feed line  9515  connected to the WiFi antenna  9503  may be 0.4 mm×5 mm with a gap of 0.2 mm to the CPW ground  19518 .  
      A full-wave simulation of the exemplary multi-band metamaterial antenna array presented in this section is illustrated in  FIG. 97 . The WiFi frequency band (2.4 GHz˜2.48 GHz) is covered by the WiFi antenna, while the WiMax frequency band (2.5 GHz˜2.7 GHz) is covered by the WiMax antenna. As further illustrated in  FIG. 97 , the return losses across the WiFi and WiMax bands can be better than −10 dB, and the isolation between the two antennas across the WiFi and WiMax bands can vary from about −17 dB to −14 dB.  
      VII.B1 Multi-Antenna, Directional Coupler System: WiFi and WiMax Antenna Array, Two-Way Directional Coupler using Microwave Coupled Line  
       FIG. 98  illustrates an example of a microwave coupled line coupler. In one implementation, the microwave coupled line coupler can be designed on a 10 mil FR-4 substrate with a dielectric constant of 4.4. The coupled line coupler can be formed by using a microstrip coupled line  9815 . As shown in  FIG. 98 , the microstrip couple line  9815  may include two transmission lines that are parallel with each other and separated by a gap, s  9810 . The microstrip coupled line  9815  impedance and the coupling level can be determined by the line width, w  9805 , and the gap width, s  9810 . Ports, P 1   9801 - 1 , P 2   9801 - 2 , P 3   9801 - 3  and P 4   9801 - 4 , of the microstrip coupled line  9815  shown in  FIG. 98  can each act as either an input port or an output port. The size of the line width and gap width may be about 0.44 mm and 0.18 mm, respectively. Based on the thickness of the substrate, dielectric constant, line width, and gap width, the coupled line coupler can be matched to 50Ω at each input and output port (P 1   9801 - 1 , P 2   9801 - 2 , P 3   9801 - 3 , P 49801 - 4 ). As previously indicated, the coupling level can be selected based on the isolation between the WiFi and WiMax antennas. For example, the length of the microstrip coupled line may be set to about 16.7 mm to achieve a maximum coupling between the input ports P 1   9801 - 1  and P 3   9801 - 3  and between the output ports P 2   9801 - 2  and P 4   9801 - 4  at about 2.52 GHz.  
      A simulation of the exemplary microwave coupled line coupler is illustrated in  FIG. 99 . The return loss result indicates that the coupler can be matched to 50Ω across a frequency range of about 2.4 GHz to 2.7 GHz. The coupling across the same bandwidth is about −16.5 dB, which is close to the average isolation between the WiFi and WiMax antennas previously presented.  
      To satisfy the phase condition for improved isolation, two 50Ω transmission lines with an additional phase delay of 46° each can be inserted between the outputs, P 2   9801 - 2  and P 4   9801 - 4  shown in  FIG. 98 , and inputs, P 1 ′  9415 - 1  and P 2 ′  9415 - 2  shown in  FIG. 94 , of the WiFi  9430  and WiMax  9425  antennas.  FIG. 100  illustrates the simulated results of the multi-band multi-antenna system shown in  FIG. 94  which may include a metamaterial WiFi antenna, a metamaterial WiMax antenna, two additional transmission lines, and a microwave coupled line coupler. Return loss and isolation shown in  FIG. 100  demonstrate that the bandwidth of return loss better than −10 dB at the WiFi and WiMax bands are retained, and the isolation between two antennas is improved. Notably, the coupling between the WiFi and WiMax antennas at frequency band edges (2.4 GHz and 2.7 GHz) is similar to the case where coupler is not included while the coupling across both bands (2.4 GHz˜2.7 GHz) is significantly reduced. Therefore, this improvement may be expected to boost the system performance.  
      VII.B2 Multi-Antenna, Directional Coupler System: WiFi and WiMax Antenna Array, Two-Way Directional Coupler Using MTM Transmission Line  
      Metamaterial technology can provide a means to design multi-antenna systems that have smaller antenna elements and allow close spacing between adjacent antennas. A MTM coupler can be constructed using a coupled metamaterial transmission line as previously mentioned. The coupled metamaterial transmission line can be constructed by placing two metamaterial transmission lines in parallel to each other where coupling may occur between the two metamaterial transmission lines. The two metamaterial transmission lines can be identical or different depending on the application requirements. The coupling between the two metamaterial transmission lines can be achieved in three ways: 1) by placing the two metamaterial transmission lines in close proximity, 2) by placing a LC-network in between two metamaterial transmission lines that are in close proximity, and 3) by placing a LC network in between two metamaterial transmission lines that are not in close proximity.  FIG. 101  illustrates an example of a MTM coupler where a one unit cell coupled metamaterial transmission line is used.  
      In another implementation, the MTM coupler can be designed on a 10 mil FR-4 substrate with a dielectric constant of 4.4. The metamaterial transmission line shown in  FIG. 101  can utilize a lumped element for (C L    10110 - 110110 - 2 , L L ,  10115 - 1   10115 - 2 ) and a microstrip line  10105  for (C R , L R ). The coupled metamaterial transmission line can be constructed by placing two identical metamaterial transmission lines in parallel and separated by a small gap. An additional lumped capacitor (C m ) can be attached between the two metamaterial transmission lines to enhance the coupling. The substrates thickness, dielectric constant, width and coupling gap of the microstrip coupled line which is realized by paralleling two microstrip lines  10105  with each other can provide a characteristic impedance of 50Ω. The width and coupling gap dimension may be about 0.44 mm and 0.21 mm, respectively. Other parameters may include the length of the microstrip line  10105 , which may be 4 mm, and C L    10110 - 110110 - 2 , L L    10115 - 1   10115 - 2 , and C m    10120 , which may be about 4 pF, 5 nH, and 0.4 pF, respectively. These values may be used to match the 50Ω impedance and the required coupling level between the two metamaterial transmission lines.  
       FIG. 102  illustrates the simulated results of the MTM coupler shown in  FIG. 101 . Notably, the return loss is better than −10 dB across the entire frequency range of about 2.4 GHz to 2.7 GHz, where the coupling level may vary from about −14.4 dB at 2.4 GHz to −13.4 dB at 2.7 GHz.  
      In another embodiment, the MTM coupler shown in  FIG. 101  may be combined with the WiFi and WiMax antennas shown in  FIGS. 95A-95F  and  FIG. 96 . In this implementation, ports P 1  ( 10101 - 1 ) and P 3  ( 10101 - 3 ) shown in  FIG. 101  can be used as input ports for input signals. The ports, P 2   10101 - 2  and P 4   10101 - 4 , as shown in  FIG. 101  can be used as the outputs of the MTM coupler. To satisfy the phase condition as previously indicated, two 50Ω transmission lines with an additional phase delay of 80° each can be inserted between the outputs of the MTM coupler, P 2   10101 - 2  and P 4   10101 - 4  shown in  FIG. 101 , and the inputs of the WiFi and WiMax antennas, P 1 ′  9415 - 1  and P 2 ′  9415 - 2  of  FIG. 94 , respectively.  
       FIG. 103  illustrates simulated results of this multi-band multi-antenna system shown in  FIG. 94  which may include a metamaterial WiFi antenna, a metamaterial WiMax antenna, two additional transmission lines and a MTM coupler. As shown in  FIG. 103 , the bandwidth having a return loss better than −10 dB at the WiFi and WiMax bands are retained while the isolation between the two antennas is improved. Notably, the coupling between the WiFi and WiMax antennas at the frequency band edges (2.4 GHz and 2.7 GHz) are similar to the case where the MTM coupler is not introduced while the coupling across both bands (2.4 GHz˜2.7 GHz) can be significantly reduced. Hence, this improvement may be expected to boost the system performance.  
      VII.C1 Multi-Antenna, Directional Coupler System: WiFi and WiMax Antenna Array, Bandpass Filters  
      In another embodiment, coupling between the WiFi and WiMax antennas can be reduced when two bandpass filters are utilized in the multi-band multi-antenna system. In another implementation, an exemplary multi-band multi-antenna system shown in  FIG. 104  may include a WiFi antenna  10405 , a WiMax antenna  10401 , a WiFi bandpass filter  10410 , and a WiMax bandpass filter  10415 . One end of the WiFi bandpass filter  10410  can be connected to the WiFi antenna  10405  to block a coupling signal radiated from the WiMax antenna  10401 . Similarly, one end of the WiMax bandpass filter  10415  can be connected to the WiMax antenna  10401  to block a signal radiated from the WiFi antenna  10405 . Thus, the isolation between the WiFi signal and the WiMax signal can be determined by the rejection strength of each bandpass filter ( 10410  and  10415 ).  
      Presently, there are various topologies of bandpass filters available. For example, a Chebyshev type of filter can be introduced to demonstrate one design concept. In one implementation, a simple lumped element method can be used to implement a bandpass filter design.  FIG. 105A  shows an example of a Chebyshev WiFi bandpass filter  10500   a . The filter shown in  FIG. 105A  may include three series capacitors ( 10520 ,  10510 ,  10515 ) and two shunt L-C resonators ( 10525 - 1  and  10530 - 1 ,  10525 - 2  and  10530 - 2 ). The three capacitors are connected in the order of C 1 L  10520 , C 2   10510 , and C 1 R  10515  where one end of each capacitor, C 1 L  10520  and C 1 R  10515 , is left unconnected. In one configuration, the unconnected end of C 1 L  10520  may be used as the bandpass filter&#39;s input while the unconnected end of C 1 R  10515  may be used as the bandpass filter&#39;s output. In yet another configuration, the unconnected end of C 1 L  10520  may be used as the output while the unconnected end of C 1 R  10515  may be used as the input. The two shunt L-C resonators can be identical and may include a shunt capacitor C 3  ( 10525 - 1 ,  10525 - 2 ) and a shunt inductor L 1  ( 10530 - 1 ,  10530 - 2 ). One shunt L-C resonator can be affixed at a connecting node A  10501  while the other shunt L-C resonator can be attached at connecting node B  10505 .  
       FIG. 105B  depicts an example of a WiMax bandpass filter  10500   b . The filter may include four series capacitors ( 10550 ,  10560 , and  10555 ) and three shunt L-C resonators ( 10580 ,  10585 ). The four capacitors can be connected in the order of C 1 L′  10550 , C 2 ′  10560 , C 2 ′  10560 , and C 1 R′  10555  where one end of each capacitor, C 1 L′  10550  and C 1 R′  10555 , is left unconnected. In one configuration, the unconnected end of C 1 L′  10550  may be used as the bandpass filter&#39;s input while the unconnected end of C 1 R′  10555  may be used as the bandpass filter&#39;s output. In yet another configuration, the unconnected end of C 1 L′  10550  may be used as the output while the unconnected end of C 1 R′  10555  may be used as the input. In the WiMax bandpass filter  10500   b , two types of shunt L-C resonators can be used: Type I  10580  and Type II  10585 . The Type I  10580  shunt L-C resonator may include a shunt capacitor C 3 ′ ( 10565 - 1 ,  10565 - 2 ) and a shunt inductor L 1 ′ ( 10575 - 1 ,  10575 - 2 ). The Type II  10585  shunt L-C resonator may include of a shunt capacitor C 4 ′  10570  and a shunt inductor L 1 ′  10575 - 2 . One Type I  10580  shunt L-C resonator can be affixed at Node C  10535 , which is in between C 1 L′  10550  and C 2 ′  10560 , while a second Type I  10580  shunt L-C resonator can be attached at Node E  10545 , which is in between C 2 ′  10560  and C 1 R′  10555 . The Type II  10585  shunt L-C resonator can be attached at Node D  10540 , which is in between the two C 2 ′  10560  capacitors.  
      For the Chebyshev WiFi bandpass filter  10500   a  shown in  FIG. 105A , values for C 1 , C 2 , C 3 , and L 1  can be designed at 0.185 pF, 0.03 pF, 0.64 pF, and 5 nH, respectively. Likewise, for the Chebyshev WiMax bandpass filter  10500   b  illustrated in  FIG. 105B , values of C 1 L′, C 1 R′, C 2 ′, C 3 ′, C 4 ′, and L 1 ′ can be designed at 0.177 pF, 0.177 pF, 0.024 pF, 0.273 pF, 0.422 pF, and 8 nH, respectively.  
       FIG. 106  illustrates the simulated results of the Chebyshev WiFi  10500   a  and WiMax bandpass filter  10500   b . The return losses for WiFi and WiMax bandpass filters ( 10500   a ,  10500   b ) are better than −10 dB across 2.4 GHz to 2.48 GHz and 2.51 GHz to 2.68 GHz, respectively. The rejection level for the WiFi bandpass filter  10500   a  at 2.5 GHz and 2.7 GHz are −2.63 dB and −23.03 dB, respectively. The rejection level for the WiMax bandpass filter  10500   b  at 2.4 GHz and 2.48 GHz are −24.48 dB and −7.83 dB.  
      The simulated results of the multi-band multi-antenna system shown in  FIG. 104  are plotted in  FIG. 107 . From  FIG. 107 , the results show that return losses of better than −10 dB for both WiFi and WiMax bands are retained.  FIG. 107  also illustrates the comparison between the isolation of the multi-antenna system shown in  FIG. 104  with and without the bandpass filters. From  FIG. 107 , the coupling between WiFi and WiMax signals decreases by integrating two bandpass filters with the WiFi and WiMax antenna array. However, this improvement is primarily at the frequency range that is close to the lower band edge portion of WiFi band and the higher band edge portion of WiMax band. Such limited improvement can be attributed to two factors: 1) a small band gap between the WiFi and WiMax bands (only 20 MHz), and 2) the higher rejection level cannot be achieved based on the presented bandpass filter type.  
      VII.C2 Multi-Antenna, Directional Coupler System: WiFi and WiMax Antenna Array, Two-Way Directional Coupler Using Microwave Coupled Line and Bandpass Filters  
      As previously indicated, the isolation between the WiFi and the WiMax antennas can be improved by using either a directional coupler or bandpass filters. Furthermore, proper operation of directional couplers may be dependent on satisfying the phase requirement. The implementation of a directional coupler in a multi-band multi-antenna system may satisfy the phase requirement and offer improved isolation but at a narrow frequency range.  
      However, the reduced frequency range may not be sufficient to cover the entire bandwidth range of 2.4 GHz to 2.7 GHz, and, thus, the implementation of the directional coupler alone may not be a sufficient solution improving the isolation between the WiFi and WiMax antennas.  
      A comparison between  FIG. 100 ,  FIG. 103  and  FIG. 107  indicates that the isolation frequency responses between the WiFi and WiMax antennas are complementary based on using the directional coupler and the bandpass filters. This suggests that integrating both the directional coupler and the bandpass filters together may be used to mitigate the drawbacks of each individual approach.  
      In yet another implementation, an exemplary multi-band multi-antenna system is presented in  FIG. 108 . The multi-band multi-antenna system shown in  FIG. 108  may include a WiFi antenna  10805 , a WiMax antenna  10801 , a directional coupler  10835 , a WiFi bandpass filter  10815 , and a WiMax filter  10820 . A WiFi signal  10825  is fed to an input of one end of the WiFi bandpass filter  10815  while a WiMax signal  10830  is fed to an input of one end of the WiMax bandpass filter  10820 . The output of the WiMax bandpass filter  10820  and the output of the WiFi bandpass filter  10815  can be connected to P 1   10810 - 1  and P 3   10810 - 3 , respectively, where P 1   10810 - 1  and P 3   10810 - 3  are inputs of the directional coupler  10835 . Outputs, P 2   10810 - 2  and P 4   10810 - 4 , of the directional coupler  10835  may be connected to the input of the WiMax antenna  10801  and the WiFi antenna  10805 , respectively. The WiFi  10815  and WiMax  10820  bandpass filters shown in  FIG. 108  are illustrated in  FIGS. 105A and 105B , respectively. The microwave coupled line coupler shown in  FIG. 98  and the MTM coupler shown in  FIG. 101  can be used for the directional coupler  10835  shown in  FIG. 108  of this embodiment.  
       FIG. 109  and  FIG. 110  illustrate simulated results of the multi-band multi-antenna system shown in  FIG. 108  that combines a microstrip coupled line coupler and a MTM coupler, respectively. Both  FIG. 109  and  FIG. 110  demonstrate that the isolation between the WiFi antenna and the WiMax antenna can be significantly reduced to less than −30 dB across the frequency range of about 2.4 GHz to 2.7 GHz. Therefore, this improvement may be expected to boost the system performance.  
      While this document contains many specifics, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination.  
      Moreover, although features may be described above are acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be exercised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.  
      Thus, particular implementations have been described. Variations and enhancements of the described implementations, and other implementations can be made based on what is described and illustrated.