Patent Publication Number: US-9887458-B2

Title: Compact butler matrix, planar two-dimensional beam-former and planar antenna comprising such a butler matrix

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to foreign French patent application No. FR 1500565, filed on Mar. 23, 2015, the disclosure of which is incorporated by reference in its entirety. 
     FIELD OF THE INVENTION 
     The present invention relates to a compact Butler matrix, a planar two-dimensional beam-former and a multiple-beam planar antenna comprising such a Butler matrix. It applies to any multiple-beam antenna, notably in the field of space applications such as satellite telecommunications, and more particularly to the antennas of small thickness. 
     BACKGROUND 
     The beam-formers are used in the multiple-beam antennas to generate output beams from input radiofrequency signals. A conventional beam-former comprises N inputs In 1  to InN, P outputs Out 1  to OutP, and a plurality of radiofrequency circuits  11 ,  12 ,  13  suitable for dividing and recombining the input radiofrequency signals according to a phase and amplitude law chosen to form output beams. There are various beam-former technologies. In  FIG. 1 , the radiofrequency circuits comprise a large number of individual waveguides  10  which cross over one another so as to allow combinations necessary for the formation of the various output beams by the radiofrequency signal combiners  12 . These beam-formers are suitable for a limited number of radiating elements and for forming a limited number of beams because they become very complex when the number of beams increases because of the necessary crossovers between the waveguides. 
     It is also known practice to form beams by using a Butler matrix consisting of a symmetrical passive circuit with N input ports and N output ports, which drives the radiating elements producing N different beams of equal amplitudes. The circuit is made up of junctions which connect the input ports to the output ports by N different and mutually parallel transmission lines  18 . There are a number of possible Butler matrix configurations. In the diagram of  FIG. 2 , the Butler matrix comprises couplers  15 , of 3 dB, 90° hybrid coupler type, making it possible to combine or divide the power of the input radiofrequency waves, phase-shifters  16  suitable for applying a phase delay of 45°, and crossover devices  17  making it possible to cross over two different transmission lines. As is known, each crossover device  17  can consist of two 3 dB, 90° couplers connected in series. An example of Butler matrix architecture with four input ports A, B, C, D and four output ports A′, B′, C′, D′ is represented in  FIG. 2 . In this example, the Butler matrix comprises four 3 dB, 90° couplers, two 45° phase-shifters and a crossover device. This type of beam-former is well suited to the formation of a small number of beams but becomes too complex when the number of beams increases. Furthermore, it allows for the formation of the beams only in a single direction of space at right angles to the transmission lines  18 . 
     According to another technology, there are planar quasi-optical beam-formers that use an electromagnetic propagation of the radiofrequency waves originating from a number of feeders placed at the input, for example feeder horns, according to a generally TEM mode of propagation between two parallel metal plates. The focusing and the collimation of the beams can be performed by an optical lens as described, for example, in the documents U.S. Pat. No. 3,170,158 and U.S. Pat. No. 5,936,588 which illustrate the case of a Rotman lens, or alternatively via a reflector as described for example in the documents FR 2944153 and FR 2 986377, the optical lens or, respectively, the reflector being inserted on the propagation path of the radiofrequency waves, between the two parallel metal plates. Different types of optical lenses can be used, these optical lenses serving essentially as phase correctors and making it possible, in most cases, to convert one or more cylindrical waves emitted by the feeds into one or more planar wave propagating in the parallel metal plate waveguide. The optical lens can comprise two opposing edges with parabolic profiles, respectively input and output. Alternatively, the optical lens can be a dielectric lens, a graded index lens with straight edges, or any other type of optical lens. In the case of a quasi-optical beam-former with optical lens, to obtain a planar antenna, it is sufficient to place input radiating elements around the input edge of the optical lens and to fix radiofrequency probes on the output edge of the optical lens, then to link each radiofrequency probe to an output radiating element via a transmission line, for example a coaxial cable. In the case of a pillbox beam-former, to obtain a planar antenna, input radiating elements are placed in front of the incorporated parabolic reflector, and output radiating elements are placed on the path of the radiofrequency waves reflected by the parabolic reflector. There are various pillbox beam-former solutions, using one or more reflectors. 
     Since this technology uses parallel plate waveguides, as an alternative to the use of a number of discrete radiating elements aligned side-by-side, it is possible to use a continuous linear aperture at the output of each parallel plate waveguide. These linear apertures, which are not spatially quantified, have performance levels very much superior to the linear networks of a number of radiating elements, for the beams that are misaligned, because of the absence of quantization, and in terms of bandwidth because of the absence of resonant propagation modes. 
     A quasi-optical beam-former is much simpler to produce than the traditional beam-formers with individual waveguides because it comprises neither coupler, nor crossover device. However, all the known planar beam-formers are capable of forming beams only in a single dimension of space, in a direction parallel to the plane of the metal plates. To form beams in two dimensions of space, in two directions, respectively parallel and orthogonal to the plane of the metal plates, it is necessary to orthogonally combine together two beam-forming assemblies, each beam-forming assembly consisting of a stacking of a number of unidirectional beam-forming layers. To orthogonally combine two beam-forming assemblies, it is further necessary to form connection interfaces, in particular input/output connectors, on each beam-forming assembly, then to link two-by-two, the various corresponding inputs and outputs of the two beam-forming assemblies by dedicated interconnecting cables as represented for example in the document U.S. Pat. No. 5,936,588 for lens-based beam-formers. This architecture is satisfactory for the formation of a small number of beams, but becomes very complex and excessively bulky when the number of beams increases. 
     To our knowledge, to date, there is no planar beam-forming device that makes it possible to form beams in two dimensions of space. Nor, moreover, are there any simple solutions for interconnecting two unidirectional beam-formers making it possible to dispense with the connection interfaces and interconnecting cables. 
     SUMMARY OF THE INVENTION 
     The aim of the invention is to remedy the drawbacks of the known beam-formers and to produce a planar two-dimensional beam-former comprising continuous transmission lines that make it possible to form beams in two dimensions of space without any connection interface or any interconnecting cable. 
     Another aim of the invention is to produce a novel Butler matrix that is particularly compact that has a novel parallel plate architecture compatible with the quasi-optical beam-formers. 
     For that, the invention relates to a compact Butler matrix comprising N waveguides, in which N is an integer number greater than three and chosen from the powers of two, couplers intended to couple two adjacent waveguides, phase-shifters and at least one crossover device suitable for crossing over two adjacent waveguides, the crossover device comprising two couplers connected in series. The Butler matrix consists of a planar multilayer structure comprising N+1 mutually parallel metal plates, stacked one on top of the other, and evenly spaced apart from one another, each space between two consecutive metal plates forming a parallel plate waveguide having two opposing walls, respectively top and bottom, consisting of the two consecutive metal plates, two adjacent metal plate waveguides comprising a common wall consisting of one of the metal plates, and the couplers, the phase-shifters and the crossover device consist of metasurfaces incorporated in the respective walls of the waveguides to be coupled, to be crossed over and to be phase-shifted. 
     Advantageously, the metasurfaces forming each coupler and the crossover device between two adjacent waveguides can consist of a metallized support provided with a plurality of through-holes evenly distributed in a coupling zone, respectively a crossover zone, of the wall common to the two corresponding adjacent waveguides, the crossover zone consisting of two coupling zones arranged cascaded one behind the other. 
     Advantageously, the metasurfaces forming each phase-shifter incorporated in a waveguide can consist of corrugations formed in a phase-shifting zone, on the two opposing walls of the corresponding waveguide. 
     Alternatively, according to a particular embodiment, each metal plate can consist of a metal coating deposited on a dielectric substrate and each coupler and crossover device between two adjacent waveguides can consist of a plurality of slits etched in the metal coating, the slits being evenly distributed throughout the coupling zone, respectively throughout the crossover zone, the crossover zone consisting of two coupling zones arranged cascaded one behind the other. 
     Alternatively, each phase-shifter can consist of a set of metal patches periodically photo-etched on the dielectric substrate of the two walls of a waveguide to be phase-shifted. 
     The invention relates also to a planar beam-former suitable for synthesizing beams in two dimensions of space, comprising at least one Butler matrix with N+1 parallel plates. 
     Advantageously, the beam-former can comprise two different Butler matrices stacked one on top of the other and respectively dedicated to two different mutually orthogonal polarizations. 
     According to an embodiment, the beam-former can further comprise N optical lenses respectively incorporated, at the output, or alternatively at the input, of the Butler matrix, in the N waveguides delimited by the N+1 metal plates. 
     Advantageously, each optical lens can be a lens of constant thickness and with graded index. 
     According to another embodiment, the beam-former can comprise two stacked stages, respectively lower and upper, each stage comprising an identical number of parallel plate waveguides, the Butler matrix being situated at the upper stage, each waveguide of the lower stage being connected in series to a waveguide of the upper stage by a respective intermediate parallel plate waveguide arranged orthogonally to the plane XOY of the two lower and upper stages, each intermediate waveguide forming a reflector incorporated in the beam-former. 
     The invention relates also to a planar antenna comprising at least one Butler matrix with N+1 parallel plates, the antenna further comprising M feeder horns connected at the input of each parallel metal plate waveguide, i.e. M.N feeder horns for the N metal plate waveguides, in which M is greater than 2, and N output feeder horns respectively connected to the N metal plate waveguides. 
     Advantageously, each output feeder horn can be a longitudinal horn coupled to a linear aperture extending transversely over the entire width of the corresponding parallel plate waveguide. 
     Advantageously, the liner apertures can be oriented in a direction at right angles to the plane of the parallel plates of the corresponding parallel plate waveguide. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other features and advantages of the invention will become clearly apparent from the rest of the description given by way of purely illustrative and nonlimiting example, with reference to the attached schematic drawings which represent: 
         FIG. 1 : a block diagram of an exemplary traditional beam-former, according to the prior art; 
         FIG. 2 : an exemplary block diagram of a Butler matrix, according to the prior art; 
         FIGS. 3 a  and 3 b   : two diagrams, respectively in perspective and in longitudinal cross section, of a first exemplary embodiment of a Butler matrix comprising a stacking of a number of parallel plate waveguides, according to the invention; 
         FIGS. 4 a  and 4 b   : two diagrams, respectively in longitudinal cross section and in plan view, illustrating an exemplary coupling zone inserted into a common metal plate between two metal plate waveguides according to the invention; 
         FIG. 5 : a diagram in longitudinal cross section, of a second exemplary embodiment of a Butler matrix comprising a composite stacking of a number of layers of etched and metallized substrates separated by spacers, according to the invention; 
         FIG. 6 : a perspective diagram, of a first exemplary two-dimensional beam-former, connected to linear apertures, and comprising a Butler matrix, according to the invention; 
         FIG. 7 : a perspective diagram of a second exemplary two-dimensional beam-former, connected to linear apertures, and comprising a Butler matrix, according to the invention; 
         FIG. 8 a   : a perspective diagram of an exemplary dielectric lens incorporated in a parallel plate waveguide, according to the invention; 
         FIG. 8 b   : a perspective diagram of an exemplary lens of constant thickness and with graded index incorporated in a parallel plate waveguide, according to the invention; 
         FIG. 9 : a diagram, in longitudinal cross section, of a third exemplary two-dimensional beam-former comprising a Butler matrix, according to the invention; 
         FIGS. 10 a  and 10 b   : a diagram, in plan view, of two stages, respectively lower and upper, of a planar antenna according to the embodiment of  FIG. 9 ; 
         FIG. 11 : a diagram in longitudinal cross section of an exemplary bi-polarization Butler matrix, according to the invention. 
     
    
    
     DETAILED DESCRIPTION 
     According to the invention, as represented in the diagrams of  FIGS. 3 a  and 3 b   , the Butler matrix consists of a planar multilayer structure comprising N+1 mutually parallel metal plates  20 , stacked one on top of the other, and evenly spaced apart from one another. The space  21  between two consecutive metal plates, consisting of air or of dielectric, forms a parallel plate waveguide PPW, the top and bottom walls of which are the two consecutive metal plates. In the various figures, the metal plates are parallel to the plane XOY, the direction X corresponding to the longitudinal direction of propagation of the radiofrequency waves in each parallel plate waveguide. Two adjacent waveguides PPW 1  and PPW 2 , PPW 2  and PPW 3 , PPW 3  and PPW 4 , comprise a common wall consisting of one of the metal plates  20 . The Butler matrix therefore comprises N parallel plate waveguides, stacked one on top of the other in the direction Z orthogonal to the plane XOY, in which N is an integer number greater than three and chosen from the powers of two. The Butler matrix also comprises couplers, for example of 3 dB, 90° hybrid coupler type, each coupler being intended to couple two adjacent waveguides together, 45° phase-shifters and crossover devices intended to mutually cross over two adjacent waveguides. According to the invention, the couplers  15 , the crossover devices  17  and the phase-shifters  16  are incorporated locally in the metal plates forming the walls of the waveguides PPW 1 , PPW 2 , PPW 3 , PPW 4  in respective coupling  22   a,    22   b,    22   c,    22   d,  crossover  24  and phase-shifting  23   a,    23   b  zones, situated on the propagation path of the radiofrequency waves and extending transversely, parallel to the direction Y, over the entire width D of the corresponding metal plate  20 . 
     In order to mutually couple or cross over two adjacent waveguides, the metal plate forming the common wall between the two adjacent waveguides comprises coupling zones and crossover zones consisting of metasurfaces locally incorporated in said common wall. A metasurface is a textured surface consisting of a dense planar distribution of small elements, identical or not, fixed, or printed, or etched on a very thin support. A metasurface is characterized by a surface impedance which locally modifies the longitudinal propagation of a wave guided in a waveguide. A metasurface has properties that are very interesting from an electromagnetic point of view because it makes it possible to control the propagation of the electromagnetic waves along its surface. Depending on the properties sought, the fixed, printed or etched elements can for example be metal blocks or metal patches or holes, or slits, evenly distributed or of variable density, the distance between two consecutive elements being less than the central operating wavelength. As represented in  FIGS. 4 a  and 4 b   , according to the invention, in each coupling zone  22   a,    22   b,    22   c,    22   d  and in the crossover zone  24  which consists of two coupling zones arranged cascaded one behind the other, the metasurface consists of a metallized support  26  provided with a plurality of through-holes  25  evenly distributed throughout the coupling zone, respectively throughout the crossover zone. The distance separating two adjacent holes is very much less, by at least a factor of three, than the wavelengths guided in the parallel plate guide. The metasurface has a high reactive surface impedance, for example 100 Ohms, the value of which depends on the density of the holes and on the length L of the coupling zone. As a nonlimiting example, at 25 GHz, a 3dB, 90° coupler synthesized by a metasurface having a reactive surface impedance of 100 Ohms has been obtained with holes evenly distributed over a length L equal to 35 mm. Two identical metasurfaces placed end to end synthesize the crossover zone. It has been verified that these surface impedances are effective for radiofrequency waves having different angles of incidence. 
     To produce a phase shift in a parallel plate waveguide, PPW 1 , PPW 4 , the two metal plates forming the top and bottom walls of the corresponding waveguide comprise phase-shifting zones  23   a,    23   b  that can consist of corrugations formed locally on the internal surface of the two corresponding metal plates and the width of which is equal to the transverse width D of the corresponding metal plates. In the example of  FIGS. 3 a  and 3 b   , the number N of waveguides is equal to four, and the number of metal plates  20  is equal to five. Between the inputs  11 ,  12 ,  13 ,  14  and the outputs O 1 , O 2 , O 3 , O 4  of the Butler matrix, a first coupling zone  22   a  is incorporated in the second metal plate common to the first waveguide PPW 1  and to the second waveguide PPW 2  and a second coupling zone  22   b  is incorporated in the fourth metal plate common to the third waveguide PPW 3  and to the fourth waveguide PPW 4 . Downstream of the two coupling zones  22   a,    22   b,  the Butler matrix comprises a crossover zone  24  consisting of two 3 dB, 90° hybrid couplers, incorporated, cascaded one behind the other, in the third metal plate common to the second and third waveguides PPW 2 , PPW 3 , and two phase-shifting zones  23   a,    23   b  respectively formed in the top and bottom walls of the first and fourth waveguides PPW 1 , PPW 4 . Finally, downstream of the phase-shifting zones  23   a ,  23   b  and of the crossover zone  24 , a third and a fourth coupling zones  23   c,    23   d,  are respectively incorporated in the second metal plate common to the first and second waveguides PPW 1 , PPW 2  and in the fourth metal plate common to the third and fourth waveguides PPW 3 , PPW 4 . In operation, in the crossover zone  24  between two adjacent waveguides PPW 2 , PPW 3 , the radiofrequency signals propagating in the two adjacent waveguides, cross over, then mutually swap over their propagation waveguide, which makes it possible to group together, two-by-two, signals which are propagated initially in non-adjacent waveguides to couple them together. Thus, in this example, the radiofrequency signals which are propagated initially in the waveguides PPW 2  and PPW 3  are swapped into the crossover zone  24  and are then propagated, downstream of the crossover zone, respectively in the waveguides PPW 3  and PPW 2 . They can therefore then be respectively coupled to the radiofrequency signals which are propagated in the waveguides PPW 4  and PPW 1 . For the Butler matrix to operate correctly for a number of incidences of radiofrequency waves propagated, according to a TEM mode, in the parallel plate waveguides, it is necessary for the phase-shifting, coupling and crossover zones to be compact and therefore for the surface impedances to be high. The size of the phase-shifting, coupling and crossover zones is all the smaller when the Butler matrix operates over a wider band and for higher radiofrequency wave incidences. 
     Alternatively, as represented in the example of  FIG. 5 , the Butler matrix can be produced according to a printed circuit technology by using a composite multilayer structure comprising a stacking of several layers consisting of etched and metallized substrates S 1 , S 2 , S 3 , S 4 , S 5  that can possibly be separated by spacers E 1 , E 2 , E 3 , E 4 . Each layer forms a waveguide comprising two mutually parallel metallized walls, each wall consisting of a metal coating  33  deposited on a dielectric substrate  32 , the spacer situated between two metallized walls being able to consist of air or comprise a material transparent to the radiofrequency waves, such as, for example, a honeycomb material, or a quartz material, or a Kevlar material, or an expanded polymer foam. The role of a spacer is to reduce the propagation losses, but this spacer is not essential. The metal coating  33  deposited on the substrate  32  is then equivalent to a metal plate  20 . The coupling  22   a,    22   b,    22   c,    22   d  and crossover  24  zones between two adjacent waveguides then consist of a plurality of slits etched in the metal coating, the slits being evenly distributed throughout the coupling zone, respectively throughout the crossover zone, the length of the crossover zone  24  being equal to twice the length of a coupling zone. The phase-shifting zones consist of metasurfaces, deposited on the metal coating, which modify the propagation delay of the radiofrequency waves. According to the invention, in the phase-shifting zone  23   a,    23   b  of a waveguide, the metasurfaces can, for example, consist of a set of metal blocks, or of metal patches  30  periodically photo-etched by photolithography on the inner face of the dielectric substrate of the two walls of the corresponding waveguide. Although this is not essential, the metal patches can for example be short-circuited by linking them to the metal coating of the wall of the corresponding waveguide, by a metallized through-hole  31  formed in the corresponding dielectric substrate. The period of distribution of the metal patches, equal to the distance between two adjacent metal patches, is less than the propagation wavelength of the radiofrequency waves in the waveguide with parallel metal walls. 
     The Butler matrix according to the invention constitutes a one-dimensional beam-former when it is used alone. According to the invention, the two-dimensional planar beam-former comprises a Butler matrix  41  comprising N parallel plate waveguides PPW, stacked one on top of the other, in which N is an integer number greater than three and chosen from the powers of two, for example, 4, 8, 16, 32, . . . , and further comprises an optical device of optical lens or reflector type. In  FIGS. 6 and 7 , the number N of waveguides PPW 1 , PPW 2 , PPW 3 , PPW 4  is equal to  4 . The structure of the Butler matrix is identical to that represented in  FIGS. 3 a  and 3 b   . Furthermore, the beam-former comprises N optical lenses  42  respectively incorporated in the N waveguides delimited by the N+1 parallel metal plates. In  FIG. 6 , the optical lenses  42  are formed in the waveguides PPW, at the input of the Butler matrix  41 , between input feeder horns  43  of each waveguide and the Butler matrix  41 , whereas, in  FIG. 7 , the optical lenses  42  are formed in the waveguides PPW at the output of the Butler matrix  41 , between the Butler matrix and output horns  44 . Each optical lens  42  can for example be a dielectric lens with a dielectric permittivity different from that of the propagation medium of the parallel plate waveguides PPW 1 , PPW 2 , PPW 3 , PPW 4  (which is equal to 1 if the waveguides PPW 1 , . . . , PPW 4  are filled with air or equal to the permittivity of the substrate  32  in the case where the waveguides consist of a stacking of layers of metallized and etched substrates). Each optical lens  42  incorporated in a parallel plate waveguide can comprise parabolic edges as represented in the waveguide PPW of  FIG. 8 a   , or be a lens of variable thickness, or, to avoid discontinuities of form, be a lens with straight edges, of constant thickness and with graded refractive index as represented in the waveguide PPW of  FIG. 8 b   , or any other type of optical lens with variable refractive index making it possible to phase-shift the radiofrequency waves according to a predefined phase law. 
     The planar beam-former that is thus produced makes it possible, with the Butler matrix  41 , to synthesize beams in the plane XOZ at right angles to the parallel plates and makes it possible, with the optical lens  42 , to synthesize beams in the plane XOY parallel to the parallel plates without any discontinuity of propagation in the parallel plate waveguides and without using any interconnection, or any link cable. 
     To obtain a planar antenna, M feeder horns  43  aligned alongside one another are connected at the input of each waveguide PPW, where M is greater than two, and at the output of the beam-former, each waveguide PPW can be linked to a number of output radiating elements or to a single longitudinal feeder horn  44  coupled to a linear aperture. In  FIGS. 6, 7, 8   a  and  8   b , the number M of feeder horns  43  is equal to 7 per waveguide, i.e. M.N input horns in total, equal to 28 for the four waveguides PPW. In  FIGS. 6 and 7 , a single longitudinal feeder horn  44  is used at the output of each waveguide PPW. Each linear aperture, coupled to the output longitudinal feeder horn  44 , extends transversely over the entire width D of the corresponding waveguide. In  FIGS. 6 and 7 , each linear aperture is oriented to radiate in a direction Z at right angles to the plane XOY of the parallel plates but this is not essential, the linear apertures could also be in the extension of the parallel plates. It should be noted that, in  FIGS. 6 and 7 , the plane of radiation of the longitudinal feeder horns is not in the extension of the parallel plates, but is folded back relative to the parallel plates. Obviously, this is not essential. It is also possible to arrange the feeder horns in the extension of the parallel plates, but in this case, it may be necessary to add a transition between each horn and the corresponding waveguide when the width of the horns is greater than the thickness of the waveguides. A longitudinal horn offers the advantage of radiating the energy over the entire aperture width of the parallel plate waveguide, which makes it possible to produce an antenna with great operating bandwidth and with a great capacity for misalignment of the beam formed and makes it possible to dispense with the array lobes. 
     The dimensions of the beam-former including optical lenses are greatly constrained by the focal distance between each optical lens  42  and the input feeder horns  43 . The greater the focal distance, the better the quality of the misaligned beams. When the optical lenses are formed at the output of the Butler matrix as represented in  FIG. 7 , the focal distance required between each optical lens and the feeder horns is advantageously used by the Butler matrix, which makes it possible to reduce the dimensions of the beam-former which is then more compact. In this embodiment, the radiofrequency waves which are propagated in the Butler matrix are no longer planar but cylindrical. 
       FIG. 9  illustrates another embodiment of a two-dimensional planar beam-former that exhibits no discontinuity of propagation. In this embodiment, the planar beam-former comprises 2 N+1 parallel plates  20  forming the respective walls of 2 N parallel plate waveguides distributed over two stages, respectively lower  50  and upper  51 . Each stage comprises N waveguides in PPW technology, stacked one on top of the other, where N is greater than three. Each parallel plate waveguide PPW 1 , PPW 2 , PPW 3 , PPW 4  of the lower stage is respectively connected in series to a parallel plate waveguide PPW 8 , PPW 7 , PPW 6 , PPW 5  of the upper stage via a respective intermediate parallel plate waveguide PPWP 1 , PPWP 2 , PPWP 3 , PPWP 4 , arranged orthogonally to the plane XOY of the two stages of the beam-former. The parallel metal plates forming the walls of each intermediate waveguide then form a reflector incorporated in the beam-former, as in a pillbox-type beam-former. The parallel metal plates forming the walls of the intermediate waveguides can comprise a profile of chosen form, which can for example be of straight form as illustrated in  FIG. 9  or of curved form, for example of parabolic form, as illustrated in  FIGS. 10 a  and 10 b   , which represent two stages, lower and upper, of a planar antenna comprising such a beam-former. At the output of the reflector, the N waveguides PPW 8 , PPW 7 , PPW 6 , PPW 5  of the upper stage are coupled together by a Butler matrix according to the invention and as described in association with  FIGS. 3 a    and  3   b.    
     To produce a planar antenna, it is then sufficient to equip each waveguide PPWP 1 , PPWP 2 , PPWP 3 , PPWP 4  of the lower stage of the beam-former with a number of feeder horns  43  and, at the output of the Butler matrix  41 , to couple each waveguide PPW 8 , PPW 7 , PPW 6 , PPW 5  of the upper stage to an output longitudinal horn  44  coupled to a linear aperture extending transversely over the entire width D of the corresponding metal plate waveguide, as represented in  FIGS. 10 a    and  10   b.    
     For operation in double polarization mode, for example circular, the invention consists in using two identical Butler matrices, respectively dedicated to each polarization, and stacked one on top of the other as represented in  FIG. 11  where each Butler matrix comprises four waveguides A, B, C, D and A′, B′, C′, D′, in parallel plate waveguide PPW technology. Since each Butler matrix is dedicated to one of the two polarizations, at the output of the beam-former, the waveguides PPW operating in a same polarization are adjacent to one another. Now, to produce an antenna with double circular polarization, it is necessary to feed output radiating elements with double circular polarization via orthomodal transducers OMT. It is therefore necessary, at the output of the Butler matrices, to group together, two-by-two, waveguides of different polarization. For that, at the output of the two Butler matrices, the invention further consists in successively crossing over adjacent waveguides chosen to group together, two-by-two, the waveguides of different polarizations. The crossovers are produced by metasurfaces incorporated in the metal plates common to two adjacent waveguides to be crossed over, as explained in relation to  FIG. 3 b   . Thus, in the example of  FIG. 11 , a first crossover is produced between the waveguides D and A′ by a metasurface incorporated in the fifth metal plate  5 . Then, two successive crossovers are respectively produced between the waveguides D and C and between the waveguides B and C by corresponding metasurfaces incorporated in the fourth and third metal plates  4 ,  3 . Symmetrically, two successive crossovers are respectively produced between the waveguides A′ and B′ and B′ and C′ by corresponding metasurfaces incorporated in the plates  6 ,  7 . The various crossovers produced make it possible, at the output of the two Butler matrices, to group together the waveguides A and A′, the waveguides B and B′, the waveguides C and C′ and the waveguides D and D′. The number of waveguides of each Butler matrix is not limited to four but must be equal to a power of two. 
     Although the invention has been described in relation to particular embodiments, it is clear that it is in no way limited thereto and that it includes all the technical equivalents of the means described as well as the combinations thereof if the latter fall within the scope of the invention.