Abstract:
A multi-level antenna array includes a plurality of patch antennas. Each layer of a plurality of layers is separated from other layers by a distance and support a portion of the plurality of patch antennas. Each of a plurality of connectors is associated with one of the plurality of layers for supplying a signal for transmission by the associated layer. A feed network on each of the plurality of layers provides a connection between a connector of the plurality of connectors associated with the layer and the portion of the plurality of patch antennas located on the layer. Each layer of the plurality of layers transmits the signal having a different orthogonal function applied thereto and multiplexes each of the signals having the different orthogonal function applied thereto onto a single transmission beam.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation application of U.S patent application Ser. No. 15/457,444, filed Mar. 13, 2017, entitled PATCH ANTENNA ARRAY FOR TRANSMISSION OF HERMITE-GAUSSIAN AND LAGUERRE GAUSSIAN BEAMS (Atty. Dkt. No. NXGN-33377), which is a continuation application of U.S. patent application Ser. No. 15/187,315, filed Jun. 20, 2016, entitled PATCH ANTENNA ARRAY FOR TRANSMISSION OF HERMITE-GAUSSIAN AND LAGUERRE GAUSSIAN BEAMS, now U.S. Pat. No. 9,595,766 issued Mar. 14, 2017 (Atty. Dkt. No. NXGN-33142), which claims priority to U.S. Provisional Application No. 62/182,227, entitled PATCH ANTENNAS FOR TRANSMISSION OF HERMITE-GAUSSIAN AND LAGUERRE-GAUS SIAN BEAMS, filed on Jun. 19, 2015 (Atty. Docket No. NXGN-32702); and which also claims priority to U.S. Provisional No. 62/233,838, entitled PATCH ANTENNAS FOR TRANSMISSION OF HERMITE-GAUSSIAN AND LAGUERRE-GAUSSIAN BEAMS, filed on Sep. 28, 2015 (Atty. Docket No. NXGN-32812); and which also claims priority to U.S. Provisional Application No. 62/242,056, entitled METHOD FOR MANUFACTURING A PATCH ANTENNA, filed on Oct. 15, 2015 (Atty. Docket No. NXGN-32844); and which also claims priority to U.S. Provisional Application No. 62/311,633, entitled HYBRID PATCH ANTENNA WITH PARABOLIC REFLECTOR, filed on Mar. 22, 2016 (Atty. Docket No. NXGN-33052), each of which is incorporated herein by reference in their entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    The present application relates to patch antennas, and more particularly to patch antennas for the transmission of Hermite-Gaussian and Laguerre-Gaussian Beams. 
       BACKGROUND 
       [0003]    When transmitting Hermite-Gaussian and Laguerre-Gaussian beams, the ability to multiplex multiple modes of these type of beams together into a single signal is needed to provide increased bandwidth. By increasing the number of Hermite-Gaussian and Laguerre-Gaussian beams that can be multiplexed together, an increased data throughput can be achieved. Thus, there is a need for antenna and transmission structures that provide for multiplexing of Hermite-Gaussian and Laguerre-Gaussian beams. 
       SUMMARY 
       [0004]    The present invention, as disclosed and described herein, in one aspect thereof, comprises a multi-level antenna array includes a plurality of patch antennas. Each layer of a plurality of layers is separated from other layers by a distance and support a portion of the plurality of patch antennas. Each of a plurality of connectors is associated with one of the plurality of layers for supplying a signal for transmission by the associated layer. A feed network on each of the plurality of layers provides a connection between a connector of the plurality of connectors associated with the layer and the portion of the plurality of patch antennas located on the layer. Each layer of the plurality of layers transmits the signal having a different orthogonal function applied thereto and multiplexes each of the signals having the different orthogonal function applied thereto onto a single transmission beam. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]    For a more complete understanding, reference is now made to the following description taken in conjunction with the accompanying Drawings in which: 
           [0006]      FIG. 1  illustrates a top view of a multilayer patch antenna array; 
           [0007]      FIG. 2  illustrates a side view of a multilayer patch antenna array; 
           [0008]      FIG. 3  illustrates a first layer of a multilayer patch antenna array; 
           [0009]      FIG. 4  illustrates a second layer of a multilayer patch antenna array; 
           [0010]      FIG. 5  illustrates a transmitter for use with a multilayer patch antenna array; 
           [0011]      FIG. 6  illustrates a multiplexed OAM signal transmitted from a multilayer patch antenna array; 
           [0012]      FIG. 7  illustrates a receiver for use with a multilayer patch antenna array; 
           [0013]      FIG. 8  illustrates a microstrip patch antenna; 
           [0014]      FIG. 9  illustrates a coordinate system for an aperture of a microstrip patch antenna; 
           [0015]      FIG. 10  illustrates a 3-D model of a single rectangular patch antenna; 
           [0016]      FIG. 11  illustrates the radiation pattern of the patch antenna of  FIG. 10 ; 
           [0017]      FIG. 12 a    illustrates the radiation pattern of a circular array for an OAM mode order 1=0; 
           [0018]      FIG. 12 b    illustrates the radiation pattern for an OAM mode order 1=0 in the vicinity of the array axis; 
           [0019]      FIG. 12 c    illustrates the radiation pattern for an OAM mode order 1=1 in the vicinity of the array axis; 
           [0020]      FIG. 12 d    illustrates the radiation pattern for an OAM mode order 1=2 in the vicinity of the array axis; 
           [0021]      FIG. 13  illustrates a multilayer patch antenna array with a parabolic reflector; 
           [0022]      FIG. 14  illustrates various configurations of the patch antenna and parabolic reflector; 
           [0023]      FIG. 15  illustrates a hybrid patch and parabolic antenna using a single reflector; 
           [0024]      FIG. 16  illustrates the simulated results of received power as a function of transmission distance with a single reflection and double reflection hybrid patch and parabolic antenna; 
           [0025]      FIG. 17  illustrates an OAM multiplexed link using hybrid patch and parabolic antenna with spiral phase plate at a receiver; 
           [0026]      FIG. 18  illustrates an OAM multiplexed link using hybrid patch and parabolic antenna for the transmitter and the receiver; 
           [0027]      FIG. 19  is a flow diagram illustrating the design and layout process of a patch antenna; 
           [0028]      FIG. 20  is a flow diagram illustrating the process for patterning a copper layer on a laminate for a patch antenna; and 
           [0029]      FIG. 21  is a flow diagram illustrating a testing process for a manufactured patch antenna. 
       
    
    
     DETAILED DESCRIPTION 
       [0030]    Referring now to the drawings, wherein like reference numbers are used herein to designate like elements throughout, the various views and embodiments of a patch antenna array for transmission of Hermite-Gaussian and Laguerre-Gaussian beams are illustrated and described, and other possible embodiments are described. The figures are not necessarily drawn to scale, and in some instances the drawings have been exaggerated and/or simplified in places for illustrative purposes only. One of ordinary skill in the art will appreciate the many possible applications and variations based on the following examples of possible embodiments. 
         [0031]      FIG. 1  illustrates a multilayer patch antenna array  102 . The multilayer patch antenna array  102  includes a first antenna layer  104  for transmitting a first ordered beam, a second antenna layer  106  for transmitting a second ordered beam and a third layer  108  for transmitting a third ordered beam. Each of the layers  104 ,  106  and  108  are stacked on a same center. While the present embodiment is illustrated with respect to a multilayer patch antenna array  102  including only three layers, it should be realized that either more or less layers may be implemented in a similar fashion as described herein. On the surface of each of the layers  104 ,  106  and  108  are placed patch antennas  110 . Each of the patch antennas are placed such that they are not obscured by the above layer. The layers  104 ,  106  and  108  are separated from each other by layer separator members  112  that provide spacing between each of the layers  104 ,  106  and  108 . The configuration of the layers of the patch antenna may be in rectangular, circular or elliptical configurations to generate Hermite-Gaussian, Laguerre-Gaussian or Ince-Gaussian beams. 
         [0032]    The patch antennas  110  used within the multilayer patch antenna array  102  are made from FR 408  (flame retardant  408 ) laminate that is manufactured by Isola Global, of Chandler Arizona and has a relative permittivity of approximately 3.75. The antenna has an overall height of 125 μm. The metal of the antenna is copper having a thickness of approximately 12 μm. The patch antenna is designed to have an operating frequency of 73 GHz and a free space wavelength of 4.1 mm. The dimensions of the input 50 Ohm line of the antenna is 280 μm while the input dimensions of the 100 Ohm line are 66 μm. 
         [0033]    Each of the patch antennas  110  are configured to transmit signals at a predetermined phase that is different from the phase of each of the other patch antenna  110  on a same layer. Thus, as further illustrated in  FIG. 3 , there are four patch antenna elements  110  included on a layer  104 . Each of the antenna elements  104  have a separate phase associated there with as indicated in  FIG. 3 . These phases include π/2, 2(π/2), 3(π/2) and 4(π/2). Similarly, as illustrated in  FIG. 4  layer  106  includes eight different patch antenna elements  110  including the phases π/2, 2(π/2), 3(π/2), 4(π/2), 5(π/2), 6(π/2), 7(π/2) and 8(π/2) as indicated. Finally, referring back to  FIG. 1 , there are included 12 patch antenna elements  110  on layer  108 . Each of these patch antenna elements  110  have a phase assigned thereto in the manner indicated in  FIG. 1 . These phases include π/2, 2(π/2), 3(π/2), 4(π/2), 5(π/2), 6(π/2), 7(π/2), 8(π/2), 9(π/2), 10(π/2), 11(π/2) and 12(π/2). 
         [0034]    Each of the antenna layers  104 ,  106  and  108  are connected to a coaxial end-launch connector  116  to feed each layer of the multilayer patch antenna array  102 . Each of the connectors  116  are connected to receive a separate signal that allows the transmission of a separate ordered antenna beam in a manner similar to that illustrated in  FIG. 2 . The emitted beams are multiplexed together by the multilayered patch antenna array  102 . The orthogonal wavefronts transmitted from each layer of the multilayered patch antenna array  102  in a spatial manner to increase capacity as each wavefront will act as an independent Eigen channel. The signals are multiplexed onto a single frequency and propagate without interference or crosstalk between the multiplexed signals. While the illustration with respect to  FIG. 2  illustrates the transmission of OAM beams at OAM 1, OAM 2 and OAM 3 ordered levels. 
         [0035]    It should be understood that other types of Hermite Gaussian and Laguerre Gaussian beams can be transmitted using the multilayer patch antenna array  102  illustrated. Hermite-Gaussian polynomials and Laguerre-Gaussian polynomials are examples of classical orthogonal polynomial sequences, which are the Eigenstates of a quantum harmonic oscillator. However, it should be understood that other signals may also be used, for example orthogonal polynomials or functions such as Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials and Chebyshev polynomials. Legendre functions, Bessel functions, prolate spheroidal functions and Ince-Gaussian functions may also be used. Q-functions are another class of functions that can be employed as a basis for orthogonal functions. 
         [0036]    The feeding network  118  illustrated on each of the layers  104 ,  106 ,  108  uses delay lines of differing lengths in order to establish the phase of each patch antenna element  110 . By configuring the phases as illustrated in  FIGS. 1-3  the OAM beams of different orders are generated and multiplexed together. 
         [0037]    Referring now to  FIG. 5 , there is illustrated a transmitter  502  for generating a multiplexed beam for transmission. As discussed previously, the multilayered patch antenna array  102  includes a connector  116  associated with each layer  104 ,  106 ,  108  of the multilayer patch antenna array  102 . Each of these connectors  116  are connected with signal generation circuitry  504 . The signal generation circuitry  504  includes, in one embodiment, a 60 GHz local oscillator  506  for generating a 60 GHz carrier signal. The signal generation circuit  504  may also work with other frequencies, such as 70/80 GHz. The 60 GHz signal is output from the local oscillator  506  to a power divider circuit  508  which separates the 60 GHz signal into three separate transmission signals. Each of these separated transmission signals are provided to an IQ mixer  510  that are each connected to one of the layer input connectors  116 . The IQ mixer circuits  510  are connected to an associated additive white gaussian noise circuit  512  for inserting a noise element into the generated transmission signal. The AWG circuit  512  may also generate SuperQAM signals for insertion in to the transmission signals. The IQ mixer  510  generates signals in a manner such as that described in U.S. patent application Ser. No. 14/323,082, filed on Jul. 3, 2014, now U.S. Patent No. 9,331,875, issued on May 3, 2016, entitled SYSTEM AND METHOD FOR COMMUNICATION USING ORBITAL ANGULAR MOMENTUM WITH MULTIPLE LAYER OVERLAY MODULATION, which is incorporated herein by reference in its entirety. 
         [0038]    Using the transmitter  502  illustrated in  FIG. 5 . A multiplexed beam (Hermite Gaussian, Laguerre Gaussian, etc.) can be generated as illustrated in  FIG. 6 . As illustrated, the multilayered patch antenna array  102  will generate a multiplexed beam  602  for transmission. In the present example, there is illustrated a multiplex OAM beam that has twists for various order OAM signals in a manner similar to that disclosed in U.S. patent application Ser. No. 14/323,082. An associated receiver detector would detect the various OAM rings  604  as illustrated each of the rings associated with a separate OAM processed signal. 
         [0039]    When signals are transmitted in free space (vacuum), the signals are transmitted as plane waves. They may be represented as described herein below. Free space comprises a nonconducting medium (σ=0) and thus J=σE=0. 
         [0040]    From experimental results Ampere&#39;s law and Faraday&#39;s law are represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     B 
                     → 
                   
                   = 
                   
                     μ 
                      
                     
                       H 
                       → 
                     
                   
                 
               
               
                 
                   
                     ∇ 
                     
                       × 
                       H 
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         D 
                       
                       
                         ∂ 
                         t 
                       
                     
                     + 
                     J 
                   
                 
               
               
                 
                   
                     Ampere 
                     &#39; 
                   
                    
                   s 
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   
                     
                       D 
                       → 
                     
                     = 
                     
                       ε 
                        
                       
                         E 
                         → 
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     J 
                     → 
                   
                   = 
                   
                     σ 
                      
                     
                       E 
                       → 
                     
                   
                 
               
               
                 
                   
                     
                       ∇ 
                       
                         × 
                         E 
                       
                     
                     = 
                     
                       
                         - 
                         
                           ∂ 
                           B 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   
                     Faraday 
                     &#39; 
                   
                    
                   s 
                 
               
             
           
         
       
     
         [0000]    If there is propagation in the z direction and therefore E and H are in the xy plane. 
         [0041]    Without the loss of any generality E may be oriented in the x-direction and H may be oriented in the y-direction thus providing propogation in the z-direction. From Ampere&#39;s-Maxwell equation, the following equations are provided: 
         [0000]    
       
         
           
             
               ∇ 
               
                 × 
                 H 
               
             
             = 
             
               
                 
                   
                     ∂ 
                     D 
                   
                   
                     ∂ 
                     t 
                   
                 
                  
                 
                     
                 
                  
                 
                   ∇ 
                   
                     × 
                     H 
                   
                 
               
               = 
               
                 
                   | 
                   
                     
                       
                         
                           x 
                           ^ 
                         
                       
                       
                         
                           y 
                           ^ 
                         
                       
                       
                         
                           z 
                           ^ 
                         
                       
                     
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             x 
                           
                         
                       
                       
                         
                           ∂ 
                           
                             ∂ 
                             y 
                           
                         
                       
                       
                         
                           ∂ 
                           
                             ∂ 
                             z 
                           
                         
                       
                     
                     
                       
                         
                           H 
                           x 
                         
                       
                       
                         
                           H 
                           y 
                         
                       
                       
                         
                           H 
                           z 
                         
                       
                     
                   
                   | 
                   
                     
 
                   
                    
                   
                     
                       
                         ( 
                         
                           
                             
                               ∂ 
                               Hz 
                             
                             
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                               y 
                             
                           
                           - 
                           
                             
                               ∂ 
                               Hy 
                             
                             
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                        
                       
                         x 
                         ^ 
                       
                     
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                               Hz 
                             
                             
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                           - 
                           
                             
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                        
                       
                         y 
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                               ∂ 
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                           - 
                           
                             
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                        
                       
                         z 
                         ^ 
                       
                     
                   
                 
                 = 
                 
                   
                     ∂ 
                     
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                       t 
                     
                   
                   ∈ 
                   E 
                 
               
             
           
         
       
     
         [0042]    Next, the vectorial wave equations may be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     ∇ 
                     
                       × 
                       H 
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         D 
                       
                       
                         ∂ 
                         t 
                       
                     
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                       ∇ 
                       
                         × 
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                     = 
                     
                       ε 
                        
                       
                         
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                           E 
                         
                         
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                           t 
                         
                       
                     
                   
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                       ∇ 
                       
                         × 
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                         - 
                         
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                           B 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   
                     ∇ 
                     
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                       - 
                       μ 
                     
                      
                     
                       
                         ∂ 
                         H 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       ∇ 
                       
                         × 
                         B 
                       
                     
                     = 
                     0 
                   
                    
                   
                       
                   
                 
               
               
                 
                   
                     
                       ∇ 
                       
                         × 
                         E 
                       
                     
                     = 
                     S 
                   
                    
                   
                       
                   
                 
               
             
           
         
       
       
         
           
             
               ∇ 
               
                 × 
                 
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                     × 
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             = 
             
               
                 
                   ∇ 
                   
                     ( 
                     
                       ∇ 
                       H 
                     
                     ) 
                   
                 
                 - 
                 
                   
                     ∇ 
                     2 
                   
                    
                   H 
                 
               
               = 
               
                 - 
                 
                   
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                     2 
                   
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                   H 
                 
               
             
           
         
       
       
         
           
             
               
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                       2 
                     
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                         ) 
                       
                     
                   
                   = 
                   
                     
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                         x 
                          
                         
                           ( 
                           
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                              
                             
                               
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                                 E 
                               
                               
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                                 t 
                               
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
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                           t 
                         
                       
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                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
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                      
                     
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                         ∂ 
                         t 
                       
                     
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                       ( 
                       
                         
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                             ∂ 
                             t 
                           
                         
                          
                         H 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
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                   2 
                 
                  
                 H 
               
               = 
               
                 
                   + 
                   εμ 
                 
                  
                 
                   
                     ∂ 
                     2 
                   
                   
                     ∂ 
                     
                       t 
                       2 
                     
                   
                 
                  
                 H 
               
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
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                   2 
                 
                  
                 H 
               
               - 
               
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                  
                 
                   
                     ∂ 
                     2 
                   
                   
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                       t 
                       2 
                     
                   
                 
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             = 
             0 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
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                                   t 
                                 
                               
                                
                               H 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           - 
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                             t 
                           
                         
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                           ( 
                           
                             ∇ 
                             
                               × 
                               H 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           - 
                           μ 
                         
                          
                         
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                             t 
                           
                         
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                           ( 
                           
                             ε 
                              
                             
                               
                                 ∂ 
                                 E 
                               
                               
                                 ∂ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
                
               
                 
 
               
               + 
               
                 
                   ∇ 
                   2 
                 
                  
                 E 
               
             
             = 
             
               
                 + 
                 με 
               
                
               
                 
                   ∂ 
                   2 
                 
                 
                   ∂ 
                   
                     t 
                     2 
                   
                 
               
                
               E 
             
           
         
       
       
         
           
             
               
                 
                   ∇ 
                   2 
                 
                  
                 E 
               
               - 
               
                 με 
                  
                 
                   
                     ∂ 
                     2 
                   
                   
                     ∂ 
                     
                       t 
                       2 
                     
                   
                 
                  
                 E 
               
             
             = 
             0 
           
         
       
     
         [0043]    Therefore in general: 
         [0000]      {right arrow over (∇)} 2   {right arrow over (E)}+{right arrow over (K)}   2   {right arrow over (E)}= 0  E ( {right arrow over (r)}, t )
 
         [0000]        {right arrow over (E)} ( r, t ) ={right arrow over (E)} ( {right arrow over (r)} ) e   −jwt   e   jkz  Propagating in z-direction 
       Therefore: 
       [0044]    
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         ∂ 
                         2 
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     + 
                     
                       
                         ∂ 
                         2 
                       
                       
                         ∂ 
                         
                           y 
                           2 
                         
                       
                     
                     + 
                     
                       
                         ∂ 
                         2 
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     E 
                     → 
                   
                    
                   
                     ( 
                     
                       r 
                       → 
                     
                     ) 
                   
                 
                  
                 
                   e 
                   
                     - 
                     jwt 
                   
                 
                  
                 
                   e 
                   jkz 
                 
               
               + 
               
                 
                   
                     W 
                     2 
                   
                   
                     y 
                     2 
                   
                 
                  
                 
                   
                     E 
                     → 
                   
                    
                   
                     ( 
                     
                       r 
                       → 
                     
                     ) 
                   
                 
                  
                 
                   e 
                   
                     - 
                     jwt 
                   
                 
                  
                 
                   e 
                   jkz 
                 
               
             
             = 
             0 
           
         
       
     
         [0000]    In free space 
         [0000]    
       
         
           
             W 
             = 
             
               
                 1 
                 
                   με 
                 
               
               = 
               
                 
                   → 
                   c 
                 
                 = 
                 
                   
                     
                       1 
                       
                         με0 
                       
                     
                      
                     
                         
                     
                      
                     
                       k 
                       2 
                     
                   
                   = 
                   
                     
                       w 
                       2 
                     
                     
                       c 
                       2 
                     
                   
                 
               
             
           
         
       
     
       Now: 
       [0045]    
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   z 
                 
               
                
               
                 
                   E 
                   → 
                 
                  
                 
                   ( 
                   
                     r 
                     → 
                   
                   ) 
                 
               
                
               
                 e 
                 jkz 
               
             
             = 
             
               
                 e 
                 jkz 
               
               [ 
               
                 
                   
                     ∂ 
                     
                       
                         E 
                         → 
                       
                        
                       
                         ( 
                         
                           r 
                           → 
                         
                         ) 
                       
                     
                   
                   
                     ∂ 
                     z 
                   
                 
                 + 
                 
                   jk 
                    
                   
                     
                       E 
                       → 
                     
                      
                     
                       ( 
                       
                         r 
                         → 
                       
                       ) 
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         z 
                       
                     
                      
                     
                       
                         E 
                         → 
                       
                        
                       
                         ( 
                         
                           r 
                           → 
                         
                         ) 
                       
                     
                      
                     
                       e 
                       jkz 
                     
                   
                   = 
                   
                     
                       
                         e 
                         jkz 
                       
                       [ 
                       
                         
                           
                             ∂ 
                             
                               
                                 E 
                                 → 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   → 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             z 
                           
                         
                         + 
                         
                           jk 
                            
                           
                             
                               E 
                               → 
                             
                              
                             
                               ( 
                               
                                 r 
                                 → 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       
                         e 
                         jkz 
                       
                       [ 
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 E 
                                 → 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   → 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               z 
                               2 
                             
                           
                         
                         + 
                         
                           jk 
                            
                           
                             
                               ∂ 
                               
                                 
                                   E 
                                   → 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     → 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         e 
                         jkz 
                       
                       [ 
                       
                         
                           jk 
                            
                           
                             
                               ∂ 
                               
                                 E 
                                 → 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                         - 
                         
                           
                             k 
                             2 
                           
                            
                           
                             
                               E 
                               → 
                             
                              
                             
                               ( 
                               
                                 r 
                                 → 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       
                         e 
                         jkz 
                       
                       [ 
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               E 
                               → 
                             
                           
                           
                             ∂ 
                             
                               z 
                               2 
                             
                           
                         
                         + 
                         
                           jk 
                            
                           
                             
                               ∂ 
                               
                                 E 
                                 → 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
       Because 
       [0046]    
       
         
           
             
               | 
               
                 2 
                  
                 k 
                  
                 
                   
                     ∂ 
                     E 
                   
                   
                     ∂ 
                     z 
                   
                 
               
               | 
             
             &gt;&gt; 
             
               | 
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     E 
                      
                     
                       ( 
                       r 
                       ) 
                     
                   
                 
                 
                   ∂ 
                   
                     z 
                     2 
                   
                 
               
               | 
             
           
         
       
     
         [0000]    Paraxial assumption 
         [0000]    
       
         
           
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     
                       E 
                       → 
                     
                      
                     
                       ( 
                       
                         r 
                         → 
                       
                       ) 
                     
                   
                 
                  
                 
                   e 
                   jkz 
                 
               
               
                 ∂ 
                 
                   z 
                   2 
                 
               
             
             = 
             
               
                 e 
                 jkz 
               
               [ 
               
                 
                   2 
                    
                   jk 
                    
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         
                           E 
                           → 
                         
                          
                         
                           ( 
                           
                             r 
                             → 
                           
                           ) 
                         
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                 
                 - 
                 
                   
                     k 
                     2 
                   
                    
                   
                     
                       E 
                       → 
                     
                      
                     
                       ( 
                       
                         r 
                         → 
                       
                       ) 
                     
                   
                 
               
               ] 
             
           
         
       
     
       Then: 
       [0047]    
       
         
           
             
               
                 ( 
                 
                   
                     
                       ∂ 
                       2 
                     
                     
                       ∂ 
                       
                         x 
                         2 
                       
                     
                   
                   + 
                   
                     
                       ∂ 
                       2 
                     
                     
                       ∂ 
                       
                         y 
                         2 
                       
                     
                   
                   + 
                   
                     2 
                      
                     jk 
                      
                     
                       
                         ∂ 
                         2 
                       
                       
                         ∂ 
                         z 
                       
                     
                   
                 
                 ) 
               
                
               
                 E 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
             = 
             0 
           
         
       
     
         [0000]    Which may be represented in cylindrical coordinates as: 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   2 
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               + 
               
                 
                   ∂ 
                   2 
                 
                 
                   ∂ 
                   
                     y 
                     2 
                   
                 
               
             
             = 
             
               
                 
                   1 
                   q 
                 
                  
                 
                   ∂ 
                   
                     ∂ 
                     q 
                   
                 
                  
                 
                   ( 
                   
                     q 
                      
                     
                       ∂ 
                       
                         ∂ 
                         q 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   1 
                   
                     q 
                     2 
                   
                 
                  
                 
                   
                     ∂ 
                     2 
                   
                   
                     ∂ 
                     
                       Φ 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0048]    This provides a paraxial wave equation in cylindrical coordinates: 
         [0000]    
       
         
           
             
               
                 
                   1 
                   q 
                 
                  
                 
                   ∂ 
                   
                     ∂ 
                     q 
                   
                 
                  
                 
                   ( 
                   
                     q 
                      
                     
                       ∂ 
                       
                         ∂ 
                         q 
                       
                     
                   
                   ) 
                 
                  
                 
                   E 
                    
                   
                     ( 
                     
                       q 
                       , 
                       Φ 
                       , 
                       z 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   1 
                   
                     q 
                     2 
                   
                 
                  
                 
                   
                     ∂ 
                     2 
                   
                   
                     ∂ 
                     
                       Φ 
                       2 
                     
                   
                 
                  
                 
                   E 
                    
                   
                     ( 
                     
                       q 
                       , 
                       Φ 
                       , 
                       z 
                     
                     ) 
                   
                 
               
               + 
               
                 2 
                  
                 jk 
                  
                 
                   
                     ∂ 
                     E 
                   
                   
                     ∂ 
                     z 
                   
                 
                  
                 
                   ( 
                   
                     q 
                     , 
                     Φ 
                     , 
                     z 
                   
                   ) 
                 
               
             
             = 
             o 
           
         
       
       
         
           
             
               P 
                
               
                 ( 
                 z 
                 ) 
               
             
             , 
             
               q 
                
               
                 ( 
                 z 
                 ) 
               
             
           
         
       
     
       Then: 
       [0049]    
       
         
           
             
               E 
               0 
             
             ∼ 
             
               e 
               
                 - 
                 
                   j 
                    
                   
                     [ 
                     
                       p 
                       + 
                       
                         
                           k 
                           
                             2 
                              
                             q 
                           
                         
                          
                         
                           ( 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
         [0050]    In general E o  can rotate on the xy-plane and the wave still propagates in the z-direction. 
         [0000]    
       
         
           
             
               
                 ∂ 
                 q 
               
               
                 ∂ 
                 z 
               
             
             = 
             1 
           
         
       
       
         
           
             
               
                 ∂ 
                 P 
               
               
                 ∂ 
                 z 
               
             
             = 
             
               - 
               
                 j 
                 q 
               
             
           
         
       
       
         
           
             q ˜Curvature of the phase front near the optical axis. 
           
         
       
     
         [0000]    
       
      
       q 
       2 
       =q 
       1 
       +z  
      
     
         [0000]    where q 2  is the output plane and q 1  is the input plane. ∞∞ 
         [0000]    
       
         
           
             
               1 
               q 
             
             = 
             
               
                 1 
                 R 
               
               - 
               
                 j 
                  
                 
                   λ 
                   
                     π 
                      
                     
                         
                     
                      
                     
                       W 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             1 
             R 
           
         
       
     
         [0000]    is the curvature of the wavefront intersecting the z-axis. 
         [0052]    Thus for a complete plane wave R=∞, the equation becomes: 
         [0000]    
       
         
           
             
               1 
               q 
             
             = 
             
               
                 1 
                 
                   R 
                   → 
                   ∞ 
                 
               
               - 
               
                 j 
                  
                 
                   λ 
                   
                     π 
                      
                     
                         
                     
                      
                     
                       W 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               q 
               0 
             
             = 
             
               
                 
                   π 
                    
                   
                       
                   
                    
                   
                     W 
                     2 
                   
                 
                 
                   
                     - 
                     j 
                   
                    
                   
                       
                   
                    
                   λ 
                 
               
               = 
               
                 
                   j 
                    
                   
                       
                   
                    
                   π 
                    
                   
                       
                   
                    
                   
                     W 
                     0 
                     2 
                   
                 
                 λ 
               
             
           
         
       
     
         [0000]    where W o  is the beam waist. 
         [0000]    
       
         
           
             q 
             = 
             
               
                 
                   q 
                   0 
                 
                 + 
                 z 
               
               = 
               
                 
                   
                     j 
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     
                       W 
                       0 
                       2 
                     
                   
                   λ 
                 
                 + 
                 z 
               
             
           
         
       
       
         
           
             
               w 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 w 
                 0 
               
                
               
                 
                   1 
                   + 
                   
                     
                       ( 
                       
                         z 
                         
                           z 
                           r 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 W 
                 2 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 W 
                 0 
                 2 
               
                
               
                 [ 
                 
                   1 
                   + 
                   
                     
                       ( 
                       
                         
                           λ 
                            
                           
                               
                           
                            
                           z 
                         
                         
                           π 
                            
                           
                               
                           
                            
                           
                             W 
                             0 
                             2 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               R 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               z 
               [ 
               
                 1 
                 + 
                 
                   
                     ( 
                     
                       
                         π 
                          
                         
                             
                         
                          
                         
                           W 
                           0 
                           2 
                         
                       
                       
                         λ 
                          
                         
                             
                         
                          
                         z 
                       
                     
                     ) 
                   
                   2 
                 
               
               ] 
             
           
         
       
       
         
           
             
               R 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               z 
               [ 
               
                 1 
                 + 
                 
                   
                     ( 
                     
                       
                         z 
                         R 
                       
                       z 
                     
                     ) 
                   
                   2 
                 
               
               ] 
             
           
         
       
       
         
           
             
               Φ 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 tan 
                 
                   - 
                   1 
                 
               
                
               
                 ( 
                 
                   z 
                   
                     z 
                     R 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             θ 
             = 
             
               λ 
               
                 π 
                  
                 
                     
                 
                  
                 
                   w 
                   0 
                 
               
             
           
         
       
       
         
           
             z 
             = 
             
               z 
               R 
             
           
         
       
       
         
           
             
               w 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 2 
               
                
               
                 w 
                 0 
               
             
           
         
       
     
         [0053]    The Rayleigh length is: 
         [0000]    
       
         
           
             
               z 
               R 
             
             = 
             
               
                 π 
                  
                 
                     
                 
                  
                 n 
               
               
                 λ 
                 0 
               
             
           
         
       
     
         [0000]    where n is the index of refraction. 
         [0000]    
       
         
           
             
               w 
               0 
               2 
             
             = 
             
               
                 w 
                 2 
               
               
                 1 
                 + 
                 
                   
                     ( 
                     
                       
                         π 
                          
                         
                             
                         
                          
                         
                           w 
                           2 
                         
                       
                       
                         λ 
                          
                         
                             
                         
                          
                         R 
                       
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             z 
             = 
             
               R 
               
                 1 
                 + 
                 
                   
                     ( 
                     
                       
                         λ 
                          
                         
                             
                         
                          
                         R 
                       
                       
                         π 
                          
                         
                             
                         
                          
                         
                           w 
                           2 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
     
         [0054]    The complex phase shift is represented by: 
         [0000]    
       
         
           
             
               j 
                
               
                   
               
                
               
                 P 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
             
             = 
             
               
                 Ln 
                  
                 
                   [ 
                   
                     1 
                     - 
                     
                       j 
                        
                       
                         ( 
                         
                           
                             λ 
                              
                             
                                 
                             
                              
                             z 
                           
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               w 
                               0 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   Ln 
                    
                   
                     
                       1 
                       + 
                       
                         
                           ( 
                           
                             
                               λ 
                                
                               
                                   
                               
                                
                               z 
                             
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 w 
                                 0 
                                 2 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
                 - 
                 
                   j 
                    
                   
                       
                   
                    
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                    
                   
                     
                       λ 
                        
                       
                           
                       
                        
                       z 
                     
                     
                       π 
                        
                       
                           
                       
                        
                       
                         w 
                         0 
                         2 
                       
                     
                   
                 
               
             
           
         
       
     
         [0055]    The real part of P(z) represents a phase shift difference between the Gaussian beam and an ideal plane wave. Thus the fundamental mode is provided: 
         [0000]    
       
         
           
             
               
                 E 
                 0 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 
                   E 
                   0 
                 
                  
                 
                   ( 
                   
                     r 
                     , 
                     z 
                   
                   ) 
                 
               
                
               
                 
                   w 
                   0 
                 
                 w 
               
                
               
                 e 
                 
                   - 
                   
                     j 
                      
                     
                       ( 
                       
                         
                           j 
                            
                           
                               
                           
                            
                           z 
                         
                         - 
                         φ 
                       
                       ) 
                     
                   
                 
               
                
               
                 e 
                 
                   - 
                   
                     
                       r 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           1 
                           
                             w 
                             2 
                           
                         
                         + 
                         
                           jk 
                           
                             2 
                              
                             R 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where: 
         [0000]    
       
         
           
             φ 
             = 
             
               
                 tan 
                 
                   - 
                   1 
                 
               
                
               
                 
                   λ 
                    
                   
                       
                   
                    
                   z 
                 
                 
                   π 
                    
                   
                       
                   
                    
                   
                     w 
                     0 
                     2 
                   
                 
               
             
           
         
       
     
         [0056]    Higher order modes may also provide other solutions. The solution of rectangular equation: 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     
                       ∂ 
                       2 
                     
                     
                       ∂ 
                       
                         x 
                         2 
                       
                     
                   
                   + 
                   
                     
                       ∂ 
                       2 
                     
                     
                       ∂ 
                       
                         y 
                         2 
                       
                     
                   
                   + 
                   
                     2 
                      
                     jk 
                      
                     
                       ∂ 
                       
                         ∂ 
                         z 
                       
                     
                   
                 
                 ) 
               
                
               
                 E 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
             = 
             0 
           
         
       
     
         [0000]    Can be determined in rectangular coordinates to be: 
         [0000]    
       
         
           
             
               E 
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 mn 
               
                
               
                 
                   C 
                   nm 
                 
                  
                 
                   E 
                   0 
                 
                  
                 
                   
                     w 
                     0 
                   
                   
                     w 
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
                  
                 
                   
                     H 
                     m 
                   
                   [ 
                   
                     
                       
                         2 
                       
                        
                       x 
                     
                     
                       w 
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                   ] 
                 
                  
                 
                   
                     H 
                     n 
                   
                   [ 
                   
                     
                       
                         2 
                       
                        
                       y 
                     
                     
                       w 
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                   ] 
                 
                  
                 
                   e 
                   
                     - 
                     
                       
                         ( 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             y 
                             2 
                           
                         
                         ) 
                       
                       
                         
                           w 
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         2 
                       
                     
                   
                 
                  
                 
                   e 
                   
                     
                       - 
                       
                         j 
                          
                         
                           ( 
                           
                             m 
                             + 
                             m 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                      
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       z 
                       
                         z 
                         0 
                       
                     
                   
                 
                  
                 
                   e 
                   
                     j 
                      
                     
                       
                         k 
                          
                         
                           ( 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                           ) 
                         
                       
                       
                         2 
                          
                         
                           R 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 z 
                 0 
               
               = 
               
                 
                   kw 
                   0 
                   2 
                 
                 2 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 w 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   w 
                   0 
                 
                  
                 
                   
                     1 
                     + 
                     
                       
                         z 
                         2 
                       
                       
                         z 
                         0 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 C 
                 60 
               
               ⇒ 
               
                 TEM 
                 OD 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 R 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   z 
                   + 
                   
                     
                       z 
                       0 
                       2 
                     
                     z 
                   
                 
                 = 
                 
                   
                     
                       
                         z 
                         0 
                         2 
                       
                       z 
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             z 
                             2 
                           
                           
                             z 
                             0 
                             2 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           z 
                           0 
                           2 
                         
                         
                           zw 
                           0 
                           2 
                         
                       
                        
                       
                         
                           w 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           kz 
                           0 
                         
                         
                           2 
                            
                           z 
                         
                       
                        
                       
                         
                           w 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0057]    The solution of cylindrical coordinates of equation: 
         [0000]    
       
         
           
             
               
                 
                   1 
                   ρ 
                 
                  
                 
                   ∂ 
                   
                     ∂ 
                     ρ 
                   
                 
                  
                 
                   ( 
                   
                     ρ 
                      
                     
                       ∂ 
                       
                         ∂ 
                         ρ 
                       
                     
                   
                   ) 
                 
                  
                 
                   E 
                    
                   
                     ( 
                     
                       ρ 
                       , 
                       ∅ 
                       , 
                       z 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   1 
                   
                     ρ 
                     2 
                   
                 
                  
                 
                   
                     
                       
                         ∂ 
                         ⋀ 
                       
                        
                       2 
                     
                      
                     
                       E 
                        
                       
                         ( 
                         
                           ρ 
                           , 
                           ∅ 
                           , 
                           z 
                         
                         ) 
                       
                     
                   
                   
                     δ∅ 
                     2 
                   
                 
               
               + 
               
                 2 
                  
                 j 
                  
                 
                     
                 
                  
                 k 
                  
                 
                   
                     2 
                      
                     
                       E 
                        
                       
                         ( 
                         
                           ρ 
                           , 
                           ∅ 
                           , 
                           z 
                         
                         ) 
                       
                     
                   
                   
                     ∂ 
                     z 
                   
                 
               
             
             = 
             0 
           
         
       
     
         [0000]    Can be determined in cylindrical coordinates to be: 
         [0000]    
       
         
           
             
               E 
                
               
                 ( 
                 
                   ρ 
                   , 
                   ∅ 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                    
                    
                   
                       
                   
                    
                   ρ 
                 
               
                
               
                 
                   C 
                   
                      
                      
                     
                         
                     
                      
                     ρ 
                   
                 
                  
                 
                   E 
                   0 
                 
                  
                 
                   
                     w 
                     0 
                   
                   
                     w 
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
                  
                 
                   
                     ( 
                     
                       
                         
                           2 
                         
                          
                         ρ 
                       
                       
                         w 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     ) 
                   
                    
                 
                  
                 
                   
                     L 
                      
                     ρ 
                   
                   ( 
                   
                     
                       
                         2 
                       
                        
                       ρ 
                     
                     
                       w 
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                   ) 
                 
                  
                 
                   e 
                   
                     - 
                     
                       
                         ρ 
                         2 
                       
                       
                         
                           w 
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         2 
                       
                     
                   
                 
                  
                 
                   e 
                   
                     
                       - 
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               ρ 
                             
                             + 
                              
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                      
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       z 
                       
                         z 
                         0 
                       
                     
                   
                 
                  
                 
                   e 
                   
                     j 
                      
                     
                         
                     
                      
                      
                      
                     
                         
                     
                      
                     ∅ 
                   
                 
                  
                 
                   e 
                   
                     j 
                      
                     
                       
                         k 
                          
                         
                             
                         
                          
                         
                           ρ 
                           2 
                         
                       
                       
                         2 
                          
                         
                           R 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0058]    The equation 
         [0000]    
       
         
           
             
               L 
                
               ρ 
             
             ( 
             
               
                 
                   2 
                 
                  
                 ρ 
               
               
                 w 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
             
             ) 
           
         
       
     
         [0000]    may also be shown as 
         [0000]    
       
         
           
             
               
                 L 
                  
                 ρ 
               
                
               
                 [ 
                 
                   
                     2 
                      
                     
                       ρ 
                       2 
                     
                   
                   
                     
                       w 
                       2 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
                 ] 
               
             
             . 
           
         
       
     
         [0059]    The lowest mode is the most important mode and in fact this transverse mode is identical for both rectangular and cylindrical coordinates. 
         [0000]    
       
         
           
             
               ϕ 
                
               
                 ( 
                 
                   l 
                   , 
                   
                     P 
                     ; 
                     z 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     2 
                      
                     P 
                   
                   + 
                   l 
                   + 
                   1 
                 
                 ) 
               
                
               
                 tan 
                 
                   - 
                   1 
                 
               
                
               
                 z 
                 
                   z 
                   0 
                 
               
             
           
         
       
       
         
           
             
               TEM 
               00 
               rect 
             
             = 
             
               TEM 
               00 
               Cyl 
             
           
         
       
       
         
           
             
               C 
               00 
             
             = 
             1 
           
         
       
       
         
           
             
               H 
               0 
             
             = 
             l 
           
         
       
       
         
           
             
               L 
               0 
               0 
             
             = 
             1 
           
         
       
     
         [0000]    then 
         [0000]    
       
         
           
             
               TEM 
               00 
             
             ⇒ 
             
               
                 E 
                  
                 
                   ( 
                   
                     ρ 
                     , 
                     z 
                   
                   ) 
                 
               
               ∼ 
               
                 
                   E 
                   0 
                 
                  
                 
                   
                     w 
                     0 
                   
                   
                     w 
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
                  
                 
                   e 
                   
                     - 
                     
                       
                         ρ 
                         2 
                       
                       
                         
                           w 
                           2 
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
                  
                 
                   e 
                   
                     
                       - 
                       jt 
                     
                      
                     
                         
                     
                      
                     
                       an 
                       
                         - 
                         1 
                       
                     
                      
                     
                       z 
                       
                         z 
                         0 
                       
                     
                   
                 
                  
                 
                   e 
                   
                     jk 
                      
                     
                       
                         ρ 
                         2 
                       
                       
                         2 
                          
                         
                           R 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0060]    Referring now to  FIG. 7 , there is illustrated a receiver  702  for demultiplexing signals received from a multiplexed signal generated using the transmitter  502  of  FIG. 5 . The receiver  702  includes a multilayer patch antenna array  102  such as that described herein above. The multilayer patch antenna array  102  receives the incoming multiplexed signal  704  and each layer  104 ,  106 ,  108  of the antenna array  102  will extract a particular order of the received multiplexed signal from each of the connector outputs  116  of a particular layer. The signals from each of the connectors  116  are applied to a mixer circuit  706  that demultiplexes the received signal in a manner similar to that discussed with respect to U.S. patent application Ser. No. 14/323,082 using a 60 GHz local oscillator signal from oscillator  708 . The demultiplexed signal may then be read using, for example, a real-time oscilloscope  710  or other signal reading device. Each of the three transmitted signals is thus decoded at the receiver  702  that were transmitted in each of the ordered OAM signals received from the transmitters  602 . In a further embodiment, a demultiplexing approach using SPP (spiral phase plate) may also be applied to detect OAM channels. 
         [0061]    The signals transmitted by the transmitter  502  or the receiver  702  may be used for the transmission of information between two locations in a variety of matters. These include there use in both front haul communications and back haul communications within a telecommunications or data network. 
         [0062]    Referring now more particularly to  FIG. 8 , there is illustrated a patch antenna element  110 . Multiple ones of these patch antenna elements  110  our located upon the multilayer patch antenna array  102  as discussed hereinabove. The antenna element  110  includes a patch  802  having a length L and a width W. The patch  802  is fed from an input transmission line  804  that is connected with the feed network  104  ( FIG. 1 ) and is resting upon a substrate  806  having a height h. The microstrip patch antenna includes a first radiating slot  808  along a first edge of the patch  802  and a second radiating slot  810  along a second edge of the patch  802 . The electronic field at the aperture of each slot can be decomposed into X and Y components as illustrated in  FIG. 9 . The Y components are out of phase and cancel out because of the half wavelength transmission line  804 . The radiating fields can be determined by treating the antenna as an aperture  900  as shown in  FIG. 9  having a width W  902  and a height h  904 . 
         [0063]    The transmission line model can be further analyzed in the following manner. G r  is the slot conductance and B r  is the slot susceptance. They may be determined according to the equations: 
         [0000]    
       
         
           
             
               G 
               r 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 W 
                                 2 
                               
                               
                                 90 
                                  
                                 
                                   λ 
                                   0 
                                   2 
                                 
                               
                             
                              
                             
                                 
                             
                              
                             for 
                              
                             
                                 
                             
                              
                             W 
                           
                           &lt; 
                           
                             λ 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               W 
                               
                                 120 
                                  
                                 
                                     
                                 
                                  
                                 
                                   λ 
                                   0 
                                 
                               
                             
                              
                             
                                 
                             
                              
                             for 
                              
                             
                                 
                             
                              
                             W 
                           
                           &gt; 
                           
                             λ 
                             0 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     B 
                     r 
                   
                 
                 = 
                 
                   
                     2 
                      
                     πΔ 
                      
                     
                         
                     
                      
                     l 
                      
                     
                       
                         ɛ 
                         eff 
                       
                     
                   
                   
                     
                       λ 
                       0 
                     
                      
                     
                       Z 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0064]    The input admittance of the patch antenna  110  can be approximated as: 
         [0000]    
       
         
           
             
               Y 
               in 
             
             = 
             
               
                 Y 
                 slot 
               
               + 
               
                 
                   Y 
                   0 
                 
                  
                 
                   
                     
                       Y 
                       slot 
                     
                     + 
                     
                       
                         jY 
                         0 
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             β 
                              
                             
                               ( 
                               
                                 L 
                                 + 
                                 
                                   2 
                                    
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   l 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       Y 
                       0 
                     
                     + 
                     
                       
                         jY 
                         slot 
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             β 
                              
                             
                               ( 
                               
                                 L 
                                 + 
                                 
                                   2 
                                    
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   l 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where Δ 1  is the end effect of the microstrip. 
         [0065]    The rectangular patch antenna  110  will resonate when the imaginary part of the input admittance goes to zero. 
         [0066]    The end effect may be calculated according to the equation: 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               l 
             
             = 
             
               0.412 
                
               
                   
               
                
               
                 h 
                  
                 
                   ( 
                   
                     
                       
                         ɛ 
                         eff 
                       
                       + 
                       0.3 
                     
                     
                       
                         ɛ 
                         eff 
                       
                       - 
                       0.258 
                     
                   
                   ) 
                 
               
                
               
                 
                   
                     ( 
                     
                       W 
                       / 
                       h 
                     
                     ) 
                   
                   + 
                   0.264 
                 
                 
                   
                     ( 
                     
                       W 
                       / 
                       h 
                     
                     ) 
                   
                   + 
                   0.8 
                 
               
             
           
         
       
       
         
           
             
               L 
               + 
               
                 2 
                  
                 Δ 
                  
                 
                     
                 
                  
                 l 
               
             
             = 
             
               
                 
                   λ 
                   g 
                 
                 2 
               
               = 
               
                 
                   λ 
                   0 
                 
                 
                   2 
                    
                   
                     
                       ɛ 
                       eff 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ɛ 
               eff 
             
             = 
             
               
                 
                   
                     ɛ 
                     r 
                   
                   + 
                   1 
                 
                 2 
               
               + 
               
                 
                   
                     
                       ɛ 
                       r 
                     
                     + 
                     1 
                   
                   2 
                 
                  
                 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           10 
                            
                           
                               
                           
                            
                           h 
                         
                         W 
                       
                     
                     ) 
                   
                   
                     - 
                     0.5 
                   
                 
               
             
           
         
       
     
         [0067]    The resonant frequency of the patch antenna  110  is given by: 
         [0000]    
       
         
           
             
               f 
               r 
             
             = 
             
               c 
               
                 2 
                  
                 
                   
                     ɛ 
                     eff 
                   
                 
                  
                 
                   ( 
                   
                     L 
                     + 
                     
                       2 
                        
                       Δ 
                        
                       
                           
                       
                        
                       l 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    Typically the width W of the aperture is given by: 
         [0000]    
       
         
           
             W 
             = 
             
               
                 c 
                 
                   2 
                    
                   
                     f 
                     r 
                   
                 
               
                
               
                 
                   ( 
                   
                     
                       
                         ɛ 
                         r 
                       
                       + 
                       1 
                     
                     2 
                   
                   ) 
                 
                 
                   
                     - 
                     1 
                   
                   / 
                   2 
                 
               
             
           
         
       
     
         [0068]    The multilayered patch antenna array  102  may transmit both Hermite Gaussian beams using the processing discussed with respect to U.S. patent application Ser. No. 14/323,082 or Laguerre Gaussian beams. When transmitting Laguerre Gaussian beams information may be transmitted in a number of fashions. A spiral phase plate and beam splitter approach may be used, a dual OAM mode antenna approach may be used or the patched antenna described herein may be utilized. These implementations would be beneficial in both fronthaul and backhaul applications. 
         [0069]    In order to transmit several OAM modes of order 1 and amplitude a 1   OAM , the antenna elements must be fed by an input signal according to the equation: 
         [0000]    
       
         
           
             
               
                 
                   a 
                   n 
                   feed 
                 
                  
                 
                   1 
                   
                     N 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       l 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       a 
                       l 
                       OAM 
                     
                      
                     
                       e 
                       
                         
                           - 
                           j 
                         
                          
                         
                             
                         
                          
                         2 
                          
                         π 
                          
                         
                           ln 
                           
                             N 
                             , 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     n 
                   
                 
               
               ∈ 
               
                 { 
                 
                   0 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     N 
                     - 
                     1 
                   
                 
                 } 
               
             
             , 
           
         
       
     
         [0070]    Note that the number of elements in the multilayer patch antenna array  102  limits the number of possible OAM modes due to sampling. Due to aliasing, modes of order greater than N/2 are actually modes of negative orders. 
         [0000]    
       
         
           
             
               
                 b 
                 
                   l 
                   ′ 
                 
                 OAM 
               
               = 
               
                 
                   
                     1 
                     
                       N 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         p 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                       
                         b 
                         p 
                         feed 
                       
                        
                       
                         e 
                         
                           j2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                             
                               pl 
                               ′ 
                             
                             
                               N 
                               , 
                             
                           
                         
                       
                        
                       
                           
                       
                        
                       p 
                     
                   
                 
                 ∈ 
                 
                   { 
                   
                     0 
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       N 
                       - 
                       1 
                     
                   
                   } 
                 
               
             
             , 
             
               
 
             
              
             
               
                 h 
                 pn 
               
               = 
               
                 β 
                  
                 
                     
                 
                  
                 
                   e 
                   
                     - 
                     
                       jkr 
                       np 
                     
                   
                 
                  
                 
                   λ 
                   
                     4 
                      
                     π 
                      
                     
                         
                     
                      
                     
                       r 
                       np 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 r 
                 pn 
               
               = 
               
                 
                   
                     D 
                     2 
                   
                   + 
                   
                     R 
                     t 
                     2 
                   
                   + 
                   
                     R 
                     r 
                     2 
                   
                   - 
                   
                     2 
                      
                     
                       R 
                       t 
                     
                      
                     
                       R 
                       r 
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           θ 
                           np 
                         
                         ) 
                       
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 θ 
                 pn 
               
               = 
               
                 2 
                  
                 
                   π 
                    
                   
                     ( 
                     
                       
                         n 
                         - 
                         P 
                       
                       N 
                     
                     ) 
                   
                 
               
             
             , 
             
               
 
             
              
             
               β 
               = 
               
                 
                   
                     g 
                     t 
                   
                    
                   
                     g 
                     r 
                   
                 
               
             
           
         
       
     
       Single Mode Link Budget 
       [0071]    
       
         
           
             
               H 
               tot 
             
             = 
             
               
                 U 
                 H 
               
                
               HU 
             
           
         
       
       
         
           
             
               b 
               OAM 
             
             = 
             
               
                 H 
                 tot 
               
                
               
                 a 
                 OAM 
               
             
           
         
       
       
         
           
             
               
                 
                   P 
                   r 
                 
                 
                   P 
                   t 
                 
               
                
               
                 ( 
                 l 
                 ) 
               
             
             = 
             
               
                 
                    
                   
                     
                       b 
                       l 
                       OAM 
                     
                     
                       a 
                       l 
                       OAM 
                     
                   
                    
                 
                 2 
               
               = 
               
                 
                    
                   
                     
                       ∑ 
                       
                         p 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           β 
                           N 
                         
                          
                         
                           e 
                           
                             
                               - 
                               jl 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               np 
                             
                           
                         
                          
                         
                           e 
                           
                             - 
                             
                               jkr 
                               np 
                             
                           
                         
                          
                         
                           λ 
                           
                             4 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               r 
                               np 
                             
                           
                         
                       
                     
                   
                    
                 
                 2 
               
             
           
         
       
     
       Asymptotic Formulation 
       [0072]    The object is to determine an asymptotic formulation of the Link budget at large distances, i.e. when D→+(∞), we seek the leading term for each value of 1 Link budget −1 are the same. 
         [0073]    The link budget is asymptotically given by: 
         [0000]    
       
         
           
             
               
                 
                   P 
                   r 
                 
                 
                   P 
                   t 
                 
               
                
               
                 ( 
                 
                    
                   l 
                    
                 
                 ) 
               
             
             = 
             
               
                  
                 
                   
                     λβ 
                     
                       4 
                        
                       π 
                        
                       
                         
                            
                           l 
                            
                         
                         ! 
                       
                     
                   
                    
                   
                     
                       ( 
                       
                         
                           
                             kR 
                             t 
                           
                            
                           
                             R 
                             r 
                           
                         
                         2 
                       
                       ) 
                     
                     
                        
                       l 
                        
                     
                   
                    
                   
                     1 
                     
                       D 
                       
                         
                            
                           l 
                            
                         
                         + 
                         1 
                       
                     
                   
                 
                  
               
               2 
             
           
         
       
     
         [0074]    From the Fraunhofer distance 2 (2max(R t R r )) 2 /λ=200λ, the link budget asymptotically tends to straight lines of slope −20 (|l|+1) dB per decade, which is consistent with an attenuation in 1/D 2|l|+2 . 
         [0000]    Asymptotic Expressions with Gains and Free Space Losses 
         [0075]    Gains and free space losses may be determined by: 
         [0000]    
       
         
           
             
               
                 
                   P 
                   r 
                 
                 
                   P 
                   t 
                 
               
                
               
                 ( 
                 
                    
                   l 
                    
                 
                 ) 
               
             
             = 
             
               
                 
                   Ng 
                   t 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
                
               
                 
                   Ng 
                   r 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
                
               
                 
                   ( 
                   
                     λ 
                     
                       4 
                        
                       π 
                        
                       
                           
                       
                        
                       D 
                     
                   
                   ) 
                 
                 
                   
                     2 
                      
                     
                        
                       l 
                        
                     
                   
                   + 
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 L 
                 
                   FS 
                   eq 
                 
               
                
               
                 ( 
                 l 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     π 
                      
                     
                         
                     
                      
                     D 
                   
                   λ 
                 
                 ) 
               
               
                 
                   2 
                    
                   
                      
                     l 
                      
                   
                 
                 + 
                 2 
               
             
           
         
       
       
         
           
             
               
                 G 
                 eq 
               
                
               
                 ( 
                 l 
                 ) 
               
             
             = 
             
               
                 Ng 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
             
           
         
       
     
         [0076]    For a fixed value of |l|, each equivalent gain increases R 2|l|  So that the link budget improves by a factor of R 4|l| . On the contrary, for a fixed value of R, when |l| increases, the link budget decreases since asymptotically the effect of D is greater than those of R t  and R r . 
         [0077]    Referring now to  FIG. 10 , there is illustrated a 3-D model of a single rectangular patch antenna designed for 2.42 GHz and only one linear polarization. The radiation pattern for this antenna is illustrated in  FIG. 11 . 
         [0078]      FIG. 12 a    illustrates the radiation patterns of the circular array for an OAM mode order 1=0 due to the higher grating lobes.  FIGS. 12 b , 12 c  and 12 d    illustrate the radiation patterns for the OAM mode orders in 1=0 ( FIG. 12 b   ), 1=1 ( FIGS. 12 c   ), and 1=2 ( FIG. 12 d   ) in the vicinity of the array axis. 
         [0079]    Asymptotic OAM path loss is illustrated by: 
         [0000]    
       
         
           
             
               
                 
                   P 
                   r 
                 
                 
                   P 
                   t 
                 
               
                
               
                 ( 
                 
                    
                   l 
                    
                 
                 ) 
               
             
             = 
             
               
                 
                   Ng 
                   t 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
                
               
                 
                   Ng 
                   r 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
                
               
                 
                   ( 
                   
                     λ 
                     
                       4 
                        
                       π 
                        
                       
                           
                       
                        
                       D 
                     
                   
                   ) 
                 
                 
                   
                     2 
                      
                     
                        
                       l 
                        
                     
                   
                   + 
                   2 
                 
               
             
           
         
       
     
         [0000]    When assuming e-band frequencies, a distance of 1000 m and a reasonable patch antenna element gains, other parameters may be calculated including the diameter for the transmitter and receiver array rings, number of antennas, etc. 
         [0080]      FIG. 13  illustrates the use of a multilevel patch antenna array in a parabolic antenna  1302 . The multilevel patch antenna array  102  is positioned at the focus point of a parabolic reflector  1306  to radiate its output signal  1304  to reflect off of the parabolic reflector  1306 . The patch antenna array  102  is mounted to the parabolic reflector  1302  via structural support members  1308 . The parabolic reflector  1306  reflects the multiplexed beam that may then be detected at some type of receiving antenna. The approach has been shown to provide a higher gain output for the antenna over one only including a multilevel patch antenna array  102  without a parabolic reflector  1306 . 
         [0081]    Referring now to  FIG. 14 , there are illustrated a number of implementations using the multilevel patch antenna array  102  and parabolic reflector  1306 . In the axial or FriendFeed implementation  1402 , the patch antenna array  102  is positioned at the focal point of the parabolic reflector  1306  by the supports  1308  to radiate signals directly into the parabolic reflector  1306  along a central axis  1404 . The reflected beam will then come straight off of the parabolic reflector  1306  and parallel to each other. If the patch antenna array is moved by an offset O form the focal point away from the parabolic reflector  1306 , the reflected beam from the parabolic reflector will focus at a particular point. In this manner by moving the array along the axis of the focus point the reflected beam can be focused at desired points along the axis. The off axis or offset feed approach  1406  positions the patch antenna array  102  off of the central axis  1404  to radiate the beam at an angle to the central axis  1404  to reflect off of the parabolic reflector  1306 . The patch antenna array  102  is held in its off axis position by support member  1308 . 
         [0082]    In the Cassegrain configuration  1408 , the multilevel patch antenna array  102  is positioned on the primary parabolic reflector  1306  and reflects outward toward a convex secondary reflector  1410  held in place by secondary reflector support members  1412 . The radiated signal reflects off of the convex reflector  1410  at an angle similar to the off axis reflection of implementation  1406  and reflects a second time off the surface of the parabolic reflector  1306 . 
         [0083]    Finally, the Gregorian implementation  1414  mounts the multilevel patch antenna array  102  on the surface of the parabolic reflector  1306  to project outward toward a concave secondary reflector  1416 . The secondary reflector  1416  is supported by secondary reflector supports  1418 . The signal radiated by the multilevel patch antenna array  102  reflects off of the secondary reflector  1416 , and a second time off of the primary parabolic reflector  1306 . Each of these cases direct the Hermite Gaussian, Laguerre Gaussian, orthogonal function multiplexed beam outward toward a receiver. 
         [0084]    The asymptotic OAM path loss using a parabolic antenna revises the previous loss equations in the following manner: 
         [0000]    
       
         
           
             
               
                 
                   P 
                   r 
                 
                 
                   P 
                   t 
                 
               
                
               
                 ( 
                 
                    
                   l 
                    
                 
                 ) 
               
             
             = 
             
               
                 
                   Ng 
                   t 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
                
               
                 
                   Ng 
                   r 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               R 
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       λ 
                       2 
                     
                   
                   ) 
                 
                 
                    
                   l 
                    
                 
               
                
               
                 
                   ( 
                   
                     λ 
                     
                       4 
                        
                       π 
                        
                       
                           
                       
                        
                       D 
                     
                   
                   ) 
                 
                 
                   
                     2 
                      
                     
                        
                       l 
                        
                     
                   
                   + 
                   2 
                 
               
                
               
                 G 
                 New 
               
             
           
         
       
     
         [0000]    The term G New  comprises the new variable arising due to the parabolic antenna. As previously discussed, assuming e-band frequencies, a distance of 1000 m and a reasonable patch antenna element gains, other parameters may be calculated including the diameter for the transmitter and receiver array rings, number of antennas, etc. 
         [0085]    The new loss equation may be further solved in the following manner: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   
                     P 
                     r 
                   
                   
                     P 
                     t 
                   
                 
                  
                 
                   ( 
                   
                      
                     l 
                      
                   
                   ) 
                 
               
               = 
               
                 
                   
                     Ng 
                     t 
                   
                   
                     
                        
                       l 
                        
                     
                     ! 
                   
                 
                  
                 
                   
                     ( 
                     
                       
                         4 
                          
                         
                           π 
                            
                           
                             ( 
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 R 
                                 t 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       
                         λ 
                         2 
                       
                     
                     ) 
                   
                   
                      
                     l 
                      
                   
                 
                  
                 
                   
                     Ng 
                     r 
                   
                   
                     
                        
                       l 
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                     ! 
                   
                 
                  
                 
                   
                     ( 
                     
                       
                         4 
                          
                         
                           π 
                            
                           
                             ( 
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 R 
                                 T 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       
                         λ 
                         2 
                       
                     
                     ) 
                   
                   
                      
                     l 
                      
                   
                 
                  
                 
                   
                     ( 
                     
                       λ 
                       
                         4 
                          
                         π 
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                         D 
                       
                     
                     ) 
                   
                   
                     
                       2 
                        
                       
                          
                         l 
                          
                       
                     
                     + 
                     2 
                   
                 
                  
                 
                   G 
                   New 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 G 
                 New 
               
               = 
               
                 
                   
                     4 
                      
                     
                       π 
                        
                       
                         ( 
                         
                           π 
                            
                           
                               
                           
                            
                           
                             R 
                             A 
                             2 
                           
                         
                         ) 
                       
                     
                   
                   
                     λ 
                     2 
                   
                 
                  
                 
                   e 
                   A 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   P 
                   r 
                 
                 
                   P 
                   t 
                 
               
                
               
                 ( 
                 
                    
                   l 
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                 ) 
               
             
             = 
             
               
                 
                   Ng 
                   t 
                 
                 
                   
                      
                     l 
                      
                   
                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
                        
                       
                         π 
                          
                         
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                             π 
                              
                             
                                 
                             
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                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
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                       2 
                     
                   
                   ) 
                 
                 
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                   l 
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                   r 
                 
                 
                   
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                   ! 
                 
               
                
               
                 
                   ( 
                   
                     
                       4 
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                         π 
                          
                         
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                             π 
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                               R 
                               r 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
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                       2 
                     
                   
                   ) 
                 
                 
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                   l 
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                   ( 
                   
                     λ 
                     
                       4 
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                       π 
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                        
                       l 
                        
                     
                   
                   + 
                   2 
                 
               
                
               
                 
                   4 
                    
                   
                     π 
                      
                     
                       ( 
                       
                         π 
                          
                         
                             
                         
                          
                         
                           R 
                           A 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 
                   λ 
                   2 
                 
               
                
               
                 e 
                 A 
               
             
           
         
       
     
         [0000]    Where R equals the radius of the parabolic antenna and e A  is the aperture efficiency of the parabolic antenna 0.55 to 0.70. 
         [0086]    Referring now to  FIG. 15 , a hybrid patch antenna  1502  and parabolic reflector  1504  uses a single reflection to generate an OAM beam. The patch antenna  1502  is placed at the focal point of the parabolic reflector  1504 . As discussed previously, the case of double reflection wherein the patch antenna  1502  is placed at the feed of a Cassegrain antenna (see  FIG. 14  reference number  1408  and  1414 ) and the generated OAM beams are reflected twice by a sub reflector and by the parabolic reflector.  FIG. 16  illustrates simulated power as a function of a transmission distance considering both the OAM beam divergence as well as the blocking affect of a patch antenna. All the power in the example illustrated in  FIG. 16  is normalized to the total power covered by the parabolic reflector. In the example illustrated in  FIG. 16 , the parabolic reflector is 4 feet in diameter. 
         [0087]    The receiver sensitivities of commercially available millimeterwave communications systems have been reviewed as listed below in Table A. The potential transmission distance of using a hybrid patch antenna and parabolic dish as the transmitter taking into consideration the commercially available receiver sensitivities is illustrated. 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Estimated distance of our approach in this 
               
               
                   
                 commercial system (transmitter power, 
               
               
                   
                 parabolic dish diameter) 
               
             
          
           
               
                   
                   
                   
                 Highest 
                 Receiver 
                 10 dBm, 
                 30 dBm, 
                 10 dBm, 
                 30 dBm, 
               
               
                 Company 
                 Model 
                 Frequency 
                 data rate 
                 sensitivity 
                 4 feet 
                 4 feet 
                 8 feet 
                 8 feet 
               
               
                   
               
               
                 Fujitsu 
                 GX4000 
                 70/80 
                 3 
                 −54 dBm 
                 1 km for 
                 1.8 km for 
                 1.8 km for 
                 &gt;3 km for 
               
               
                   
                   
                 GHz 
                 Gbps 
                   
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
               
               
                   
                   
                   
                   
                   
                 400 m for 
                 700 m for 
                 1.2 km for 
                 2 km for 
               
               
                   
                   
                   
                   
                   
                 OAM 2 
                 OAM 2 
                 OAM 2 
                 OAM 2 
               
               
                 E-band 
                 E-Link 
                 70/80 
                 1.25 
                 −66 dBm 
                 1.4 km for 
                 2.5 km for 
                 2.5 km for 
                 &gt;3 km for 
               
               
                 communi- 
                 1000Q 
                 GHz 
                 Gbps 
                   
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
               
               
                 cations 
                   
                   
                   
                   
                 500 m for 
                 800 m for 
                 1.6 km for 
                 2.5 km for 
               
               
                   
                   
                   
                   
                   
                 OAM 2 
                 OAM 2 
                 OAM 2 
                 OAM 2 
               
               
                   
                 E-Link 
                 70/80 
                 3 
                 −48 dBm 
                 800 m for 
                 1.5 km for 
                 1.5 km for 
                 3 km for 
               
               
                   
                 Eagle 
                 GHz 
                 Gbps 
                 (estimated) 
                 OAM1, 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
               
               
                   
                   
                   
                   
                   
                 350 m for 
                 500 m for 
                 1 km for 
                 1.7 km for 
               
               
                   
                   
                   
                   
                   
                 OAM2 
                 OAM 2 
                 OAM 2 
                 OAM 2 
               
               
                 Bridge 
                 GE60 
                 60 
                 1 
                 −60 dBm 
                 1.2 km for 
                 2.2 km for 
                 2.2 km for 
                 &gt;3 km for 
               
               
                 Wave 
                   
                 GHz 
                 Gbps 
                 (estimated) 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
               
               
                   
                   
                   
                   
                   
                 400 m for 
                 700 m for 
                 1.5 km for 
                 2.2 km for 
               
               
                   
                   
                   
                   
                   
                 OAM 2 
                 OAM 2 
                 OAM 2 
                 OAM 2 
               
               
                   
                 AR60 
                 60 
                 1 
                 −60 dBm 
                 1.2 km for 
                 2.2 km for 
                 2.2 km for 
                 &gt;3 km for 
               
               
                   
                   
                 GHz 
                 Gbps 
                 (estimated) 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
                 OAM 1, 
               
               
                   
                   
                   
                   
                   
                 400 m for 
                 700 m for 
                 1.5 km for 
                 2.2 km for 
               
               
                   
                   
                   
                   
                   
                 OAM 2 
                 OAM 2 
                 OAM 2 
                 OAM 2 
               
               
                   
                 AR60X 
                 60 
                 1 
                 −70 dBm 
                 1.6 km for 
                 3 km for 
                 &gt;3 km for 
                 &gt;3 km for 
               
               
                   
                   
                 GHz 
                 Gbps 
                 (estimated) 
                 OAM 1, 
                 OAM1, 
                 OAM 1, 
                 OAM 1, 
               
               
                   
                   
                   
                   
                   
                 600 m for 
                 900 m for 
                 1.8 km for 
                 3 km for 
               
               
                   
                   
                   
                   
                   
                 OAM 2 
                 OAM 2 
                 OAM 2 
                 OAM 2 
               
               
                   
               
             
          
         
       
     
         [0088]      FIG. 17  illustrates a configuration for an OAM multiplexed 60 GHz link using a hybrid patch and parabolic antenna  1702 . The receivers  1704  are configured with spiral phase plates  1706  and focusing lens  1708 . The spiral phase plate  1706  demultiplexes the received OAM signals.  FIG. 18  illustrates another hybrid patch parabolic antenna configuration that includes a hybrid patch and parabolic antenna for both the transmitter  1802  and receiver  1804  with no beam splitters. The configurations of  FIGS. 17 and 18  will use 60 GHz millimeter wave data links to multiplex OAM +1 and OAM −2 using the hybrid patch and parabolic antennas. The system using the hybrid patch and parabolic antennas will provide kilometer transmission distances. 
         [0089]    The production of the patch antennas  110  are carried out through a design and layout process as generally illustrated in  FIG. 19 , a clean room procedure for production of the antenna as generally illustrated in  FIG. 20  and a final testing process as illustrated in  FIG. 21 . Referring now to  FIG. 19 , the design and layout process is more particularly described. Initially, the patch antenna is designed and simulated at step  1902  using ANSYS HFSS with a microstrip feed structure. ANSYS HFSS comprises a high-frequency structural simulator. The software within the device stimulates 3-D full wave electromagnetic field. The ANSYS HFSS creates a GDSII file (graphic database system file used to control integrated circuit photomask plotting) from the HFSS simulation and exports the GDSII file to an AWR (Applied Wave Research Corporation) Microwave Office (MWO) layout at step  1904 . In order to measure the antenna with ground signal ground probe feeding, a previously design conductor backed coplanar waveguide to microstrip transition design that has been fabricated using Agilent Momentum is also imported at step  1906  as a GDSII Agilent Momentum file into the AWR MWO Layout. The two designs are brought together at step  1908  and a weight and etch compensation of 12 μm is added to the lateral dimensions to account for isotropic wet etch used in the fabrication process. The final GDSII file for the layout is exported at step  1910  and provided to a clean room for fabrication at step  1912 . 
         [0090]    Referring now to  FIG. 20 , there is illustrated the clean room process for patterning a copper layer on the FR408 laminate. Initially, the double-sided Cu FR408 laminate is cut using scissors at step  2002  to an appropriate size (typically 1.5″×1.5″). The FR408 laminate is cleaned by rinsing the laminate at step  2004  with acetone, isopropanol (IPA) and nitrogen (N 2 ) and dried in a solvent hood or using program 2 of a CPK Solvent Spinner with the appropriate chuck. The laminate is dehydrate baked at 130° C. for two minutes on a hot plate (for example, a Cole Parmer digital hotplate) at step  2006 . Next, at step  2008 , hexamethyldisilizane (HMDS) is deposited on the laminate by a rain method using a Yield Engineering YES— 310  vacuum hood oven. The laminate samples are placed in the HMDS oven for 20 minutes to improve resist adhesion. Next, at step  2010 , the mask is cleaned using program 2 of a CPK Solvent Spinner with the appropriate chuck. The mask is further cleaned using an automated mask cleaner (Ultratech Mask Cleaner) using program 0 DIW only at step  2012 . 
         [0091]    The lithography process is performed at steps  2014 - 2034 . First, Shipley S 1813  photoresist is spun on to the backside of the laminate at step  2014  to protect the ground layer using for example a Brewer Science Cee Spin Coater System. In one embodiment, the spin coater system will operate at 3000 rpm with 3000 rpm/s for 60 seconds. The sample is soft baked at step  2016  at 115° C. for 90 seconds on a hot plate and hard baked at step  2018  at 130° C. for 60 seconds on the hotplate. S 1813  resist is spun onto the top side pattern copper layer at step  2022 . In one embodiment, this is carried out at 3000 rpm with 3000 rpm/s for 60 seconds. The sample is soft baked at 115° C. for 90 seconds on a hot plate at step 2022. The top side of the sample is exposed at step  2024  with 110 mJ/cm2 using Karl Suss MA6 BA6 Contact Aligner/Printer. Next, the circuit is developed at step  2026  with Microposit MF-319 for 60 seconds in a beaker. The sample is rinsed with deionized water (DIW) and N 2  in a base hood. A reactive ion etching process is performed at step  2032  to remove excess photoresist using Techniques Series 85 RIE. This is achieved by applying O 2  only at 180 mTorr with 50 W for 15 seconds. The sample is hard baked at step  2034  at 130° C. for 60 seconds on a hot plate. The lithography is checked at step  2036  under a Leica Inm Optical microscope to make sure the lithography is correct and that the gaps are defined and not overdeveloped. 
         [0092]    The 12 μm copper layer is etched at steps  2038 - 2046 . The copper is etched in one minute intervals at step  2038  by agitating the sample in a Cu etchant. Inquiry step  2040  determines if the Cu etching process is complete, and if not, the sample is rotated at step  2042  by 90° and returns to agitate the sample within the Cu etchant at step  2038 . When inquiry step  2040  determines that the Cu etching process is completed control passes to step  2044  wherein the sample is rinsed with DIW and N 2  and dried within a base hood. The sample is checked at inquiry step  2046  using a microscope to determine if the Cu has been completely removed. If not, control passes back to step  2038  for further agitation within the Cu etchant. If all of the Cu has been removed control passes to the stripping of the photoresist process. 
         [0093]    The stripping of the photoresist occurs by first rinsing the sample with acetone, IPA, DIW and N 2  and drying within a solvent hood or using program 2 in CPK Solvent Spinner with the appropriate chuck. The sample is dehydrate baked at step  2050  at 130° C. for five minutes on a hot plate. The etched laminate samples are examined at step  2052  under a microscope to make sure that gaps are etched with no over etching of areas within the sample. 
         [0094]    The created patch antenna may be tested as illustrated in  FIG. 21  to confirm operation of the antenna. Initially, at step  2102 , a DC test is performed upon the antenna to make sure that the G-S-G feed is not shorted. An RF test is performed at step  2104  to measure the S 11 -Return Loss across the frequency bands using Agilent VNA on Cascade M150 probe station. The radiation pattern of the antenna may then be measured at step  2106  at the appropriate frequencies using a NSI spherical near field scanner. 
         [0095]    It will be appreciated by those skilled in the art having the benefit of this disclosure that this patch antenna array for transmission of Hermite-Gaussian and Laguerre-Gaussian beams provides for the transmission of multiplexed Hermite Gaussian and Laguerre Gaussian modes in a single transmission beam. It should be understood that the drawings and detailed description herein are to be regarded in an illustrative rather than a restrictive manner, and are not intended to be limiting to the particular forms and examples disclosed. On the contrary, included are any further modifications, changes, rearrangements, substitutions, alternatives, design choices, and embodiments apparent to those of ordinary skill in the art, without departing from the spirit and scope hereof, as defined by the following claims. Thus, it is intended that the following claims be interpreted to embrace all such further modifications, changes, rearrangements, substitutions, alternatives, design choices, and embodiments.