Compact dual-band orthomode transducer with linear polarisation

EP4677690A1Pending Publication Date: 2026-01-14SWISSTO 12 SA
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Patent Information

Application Number
EP2024712298
Authority / Receiving Office
EP · EP
Patent Type
Applications
Current Assignee / Owner
Priority Date
2023-03-10
Filing Date
2024-03-07
Publication Date
2026-01-14

AI Technical Summary

Technical Problem

Traditional orthomode transducers face challenges in being compact while maintaining wide transmission/reception bandwidth, with most being either compact but not broadband or broadband but bulky, limiting their use in dense antenna networks.

Method used

A compact dual-band orthomode transducer with linear polarization is achieved through a hybrid coupler and septum polarizer arrangement, utilizing additive manufacturing to minimize bulk and footprint, with waveguides arranged symmetrically and a coupling portion that reduces overhanging portions for efficient manufacturing.

Benefits of technology

The solution results in a compact, wideband orthomode transducer with reduced bulk, enabling denser antenna networks and efficient operation across multiple frequency bands.

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Abstract

The present invention relates to a broadband orthomode transducer with linear polarisation, obtained by additive manufacturing, comprising: a hybrid coupler comprising: a first waveguide comprising a first input port and a first single-polarisation output port; and a second waveguide, comprising a second input port and a second single-polarisation output port; and a coupling portion connecting the first waveguide to the second waveguide; a septum polariser comprising: a third single-polarisation input port connected to the first output port; and a fourth single-polarisation input port connected to the second output port; and a third single-polarisation output port; wherein the first and second waveguides are arranged symmetrically with respect to a plane of symmetry containing a septum of the septum polariser.
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Description

Compact dual-band orthomode transducer with linear polarization Technical field

[0001] The present invention relates to a linearly polarized orthomode transducer which is both compact and dual-band. State of the art

[0002] Orthomode transducers (abbreviated as TOM) are passive components widely used in radio frequency antennas to enable their operation in both reception and transmission.

[0003] Since their use is particularly widespread in antennas on board telecommunications satellites, limiting the weight and size of such antennas is a crucial issue. Furthermore, the ability of an antenna (and therefore of a TOM) to transmit / receive on several frequency bands is also a determining factor.

[0004] However, traditional TOMs generally struggle to be compact while ensuring a wide transmission / reception bandwidth. Indeed, on the one hand, there are TOMs, such as side-arm TOMs, which are compact but not wideband. In some cases, a side arm can be extended to operate in a second frequency band, however this extension is only possible for one of the two polarizations. On the other hand, there are TOMs, such as "Boifot" junctions or "Turnstile" junctions, which are wideband, but which are not compact since they have an aperture greater than the wavelength X of a wave having the highest operating frequency of the TOM.

[0005] Document EP2330681 A1 describes a single-band orthomode transducer comprising a septum polarizer whose septum allows a phase shift of 180° so as to produce a 45° polarization.

[0006] Document US2012 / 0319799A1 describes an orthomode transducer comprising a first septum polarizer connected to a second corrugated polarizer for undoing the circular polarization generated by the first polarizer and producing a 45° polarization. A wider performance band is theoretically possible if the length of the transducer is increased, which increases its footprint.

[0007] EP2047564B1 describes an orthomode transducer comprising a coupling portion in the form of a side arm coupled to the transducer via a slot, allowing a 90° rotation of the side waveguide. The side arm increases the footprint of the transducer. Furthermore, broadband performance is not demonstrated for this type of transducer. Brief summary of the invention

[0008] An object of the present invention is to provide an orthomode transducer free from the limitations of known orthomode transducers.

[0009] Another aim of the invention is to propose a wideband orthomode transducer with reduced bulk.

[0010] These aims are achieved by means of the subject matter of the claims and in particular by means of a linearly polarized, broadband orthomode transducer obtained by additive manufacturing comprising: a hybrid coupler comprising: a first waveguide comprising a first input port and a first single-polarized output port; and a second waveguide, comprising a second input port and a second single-polarized output port; and a coupling portion connecting the first waveguide to the second waveguide; the orthomode transducer further comprising: a septum polarizer comprising: a third single-polarized input port connected to the first output port; and a fourth single-polarized input port connected to the second output port; and a third single-polarized output port;wherein the first and second waveguides are arranged symmetrically about a plane of symmetry containing a septum of the septum polarizer.;

[0011] The symmetrical arrangement of the first and second waveguides of the coupler makes it possible to limit the footprint of the transducer, i.e. to obtain a compact cross-section over the entire length of the transducer. This results in a reduced footprint, allowing, for example, the densification of such devices in an antenna array.

[0012] According to a first embodiment, the coupling portion connects a first lateral face of the first waveguide and a second lateral face of the second waveguide, the first and second lateral faces being arranged in the same plane.

[0013] The first side face may correspond to a face of the first minimum-dimension waveguide, and the second side face corresponds to a face of the second minimum-dimension waveguide.

[0014] Advantageously, this configuration makes it possible to further reduce the size by arranging the two waveguides of the coupler so that the larger lateral faces of these waveguides are opposite each other. Thus, the footprint of the coupler is reduced.

[0015] According to one embodiment, the coupling portion comprises a prism with a trapezoidal base, a first portion of a rectangular face of the prism being in contact with the first lateral face of the first waveguide and a second portion of the rectangular face of the prism being in contact with the second lateral face of the second waveguide.

[0016] This geometry of the coupling portion makes it easier to manufacture the coupler additively by limiting the overhanging portions.

[0017] The coupling portion may include an impedance matching element.

[0018] According to a second embodiment, the coupling portion connects a first lateral face of the first waveguide and a second lateral face of the second waveguide, the first and second lateral faces being arranged in separate and parallel planes.

[0019] This arrangement advantageously makes it possible to limit the footprint of the transducer by placing the coupling portion between the two waveguides. Indeed, the outer faces of the coupler, that is to say the faces of one waveguide not directly facing a face of the other waveguide, do not include a protruding element which would increase the footprint of the transducer.

[0020] Advantageously, the coupling portion may comprise a plurality of branches, a first end of each branch being connected to the first waveguide and a second end of each branch being connected to the second waveguide.

[0021] According to one embodiment, each branch forms a two-sided roof, a first roof section being adjacent to the first lateral face and a second roof section being adjacent to the second lateral face. This geometry of the branches makes it possible to facilitate additive manufacturing by limiting the overhanging portions.

[0022] A roof edge formed by the junction of the first and second sides may be contained in a plane perpendicular to a printing direction.

[0023] In order to facilitate additive manufacturing, the first pan may form an angle with the first side face of the first waveguide of between 35° and 55° and the second pan may form an angle with the second side face of the second waveguide of between 35° and 55°.

[0024] In order to further limit the sections of the cantilevered coupling portion, each branch may form a double gable roof comprising a first gable roof of which a first gable roof is adjacent to the first side face and a second gable roof is adjacent to the second side face, and comprising a second gable roof, of which a third gable roof is adjacent to the first side face and a fourth gable roof is adjacent to the second side face.

[0025] The first side and the third side can advantageously form an angle with the first lateral face of between 35° and 55° and the second and the fourth side can form an angle with the second lateral face of between 35° and 55°.

[0026] An edge of the first roof formed by the junction of the first and second panes may form an angle with a printing direction of between 35° and 55° and an edge of the second roof formed by the junction of the third pane and the fourth pane may form an angle with the printing direction of between 35° and 55°. Brief description of the figures

[0027] Examples of implementation of the invention are indicated in the description illustrated by the appended figures in which: • Figure 1 schematically illustrates an orthomode transducer according to the invention. • Figure 2 illustrates a septum polarizer. • Figure 3 illustrates a hybrid coupler comprising a trapezoidal coupling portion. • Figures 4a and 4b illustrate an orthomode transducer incorporating the hybrid coupler of Figure 3. • Figures 5a and 5b illustrate a hybrid coupler whose coupling portion comprises a plurality of branches suitable for additive manufacturing. • Figures 6a and 6b illustrate a hybrid coupler whose coupling portion comprises a plurality of branches suitable for additive manufacturing. • Figures 7a and 7b illustrate an orthomode transducer including the hybrid coupler shown in Figures 6a and 6b. Example(s) of embodiment of the invention

[0028] As illustrated in Figure 1, the orthomode transducer of the present invention comprises a hybrid coupler 1 connected to a septum polarizer 2. The hybrid coupler 1 comprises two waveguides (10, 11) arranged parallel and connected to each other by a coupling portion 12. The septum polarizer 2 is connected to the output ports of the hybrid coupler 1 via two input ports.

[0029] When a radiofrequency device including the present orthomode transducer operates in transmission, an electromagnetic wave is propagated from the hybrid coupler 1 to the septum polarizer 2, and when it operates in reception, from the septum polarizer to the hybrid coupler. The terminology used in the context of the present invention, such as "input port" or "output port", corresponds to a transmission mode of operation although the transducer can operate indifferently in transmission and / or reception.

[0030] The direction of propagation of the waves in the transducer is therefore parallel to the longitudinal direction of the waveguides of coupler 1 and polarizer 2.

[0031] The orthomode transducer of the present invention is obtained by additive manufacturing. The expression "additive manufacturing" describes any method of manufacturing parts by adding material, according to computer data stored on a computer medium and defining a model of the part. In addition to stereolithography, the expression also designates other manufacturing methods by hardening or coagulation of liquid or powder in particular, including without limitation methods based on ink jets (binder jetting), DED (Direct Energy Deposition), EBFF (Electron beam freeform fabrication), FDM (fused deposition modeling), PFF (plastic freeforming), by aerosols, BPM (ballistic particle manufacturing), powder bed, SLS (Selective Laser Sintering), ALM (additive Layer Manufacturing), polyjet, EBM (electron beam melting), photopolymerization, etc.

[0032] Thus, although the orthomode transducer comprises two elements with different functions, i.e. a hybrid coupler and a septum polarizer, its manufacture does not require any assembly after the 3D printing steps. The coupler and the polarizer are therefore typically made from a single piece.

[0033] Hybrid couplers are four-port directional couplers used to separate or combine waves with particular phase relationships. There are mainly two types of hybrid couplers, one producing a 90° phase shift between the two output ports and the other producing a 180° phase shift between the two output ports. Hybrid couplers also function as power dividers since generally the wave undergoes an attenuation of 3dB, that is to say that the waves propagated by each of the output ports have a power equal to 50% of the power of the input wave.

[0034] Figure 2 illustrates a prior art Riblet-type hybrid coupler consisting of two waveguides coupled together through an opening in the adjoining walls of the two waveguides.

[0035] As schematically illustrated in Figure 1, the first waveguide 10 thus comprises a first input port 100 and a first output port 101, and the second waveguide comprises a second input port 110 and a second output port 111. The phase shift at the output ports depends on the coupling portion 12. Each of these four ports is single-polarized.

[0036] The coupling portion 12 of the coupler 1 allows a wave propagating in the first waveguide 10 to pass, at least partially, into the second waveguide 11 and a wave propagating in the second waveguide 11 to pass, at least partially, into the first waveguide 10.

[0037] Thus, by way of example, a wave propagating through the first input port 100 of the first waveguide 10 will be distributed between the first and second waveguides (10, 11) via the coupling portion 12 and thus exit the hybrid coupler 1 through the first and second output ports (100, 101).

[0038] The output ports (101, 111) of the hybrid coupler are connected to the septum polarizer 2 via a third input port 200 and a fourth input port 210 of the polarizer, respectively. The other end of the polarizer 2 comprises a third output port 201. The third output port may be connected to a radiating element of the antenna, or even function as a radiating element itself in certain embodiments. Each of the third and fourth input ports (200, 210) of the polarizer are single-polarized. The third output port 201 of the polarizer is dual-polarized.

[0039] Although the various embodiments shown in the figures illustrate waveguides with rectangular sections, the invention is not limited to these geometries. Indeed, the first and second waveguides (10, 11) as well as the septum polarizer 2 may also have a circular, elliptical, polygonal (regular or irregular) section, e.g. triangular, pentagonal, hexagonal, octagonal, etc.

[0040] The hybrid coupler 1 is used to excite the third and fourth input ports of the polarizer 2 simultaneously so as to create two circular polarizations. The combination of these two circular polarizations by the septum of the polarizer results in a linear polarization in the third output port 201.

[0041] The septum 22 of the polarizer 2 is arranged in a plane parallel to the direction of propagation of the waves in the transducer. The septum is typically of variable height, the highest portion of the septum being arranged at the end of the polarizer comprising the third and fourth input ports (200, 210). The height of the septum may typically decrease linearly or in steps. Figure 2 illustrates a polarizer 2 comprising a septum 22 decreasing in steps.

[0042] The orthomode transducer of the present invention can be implemented in different radio frequency devices for various frequency bands depending on their application. The present invention can be typically implemented in devices for the bands: X, Ku, Ka, QV, Ku / Ka, Ka / QV.

[0043] Advantageously, the first and second waveguides (10, 11) of the hybrid coupler 1 are arranged symmetrically with respect to a plane of symmetry containing the septum 22 of the polarizer 2.

[0044] Advantageously, the section of the septum polarizer 2 measured perpendicular to the direction of propagation of the waves in the orthomode transducer is the same as the section of the hybrid coupler 1 measured perpendicular to this same direction of propagation. Thus, the diameter of the orthomode transducer (or transducer footprint) measured in a plane perpendicular to the direction of propagation is essentially constant along the direction of propagation. This characteristic allows to increase the compactness of the transducer and therefore reduce its size, particularly for use in a compact antenna array.

[0045] As mentioned above, depending on the coupling portion 12, the hybrid coupler 1 produces a phase shift of 90° or 180°. This means that, in transmission, an electromagnetic wave propagating through the first or second input port (100, 110) of the hybrid coupler will be divided into two waves phase-shifted relative to each other by 90° or 180° between the first and second output ports (101, 111). Each of the two output waves is further attenuated by -3dB relative to the input wave. In reception, the hybrid coupler receives two waves phase-shifted by 90° or 180° and combines them into a wave whose power is doubled, i.e. increased by 3dB. Two main embodiments are described below, each corresponding to one of the two phase shifts.

[0046] A hybrid coupler 1 according to a first embodiment is illustrated in FIG. 3. The first and second waveguides 10, 11 are connected by the coupling portion 12.

[0047] In transmission, this coupler divides a wave propagated by the first or second input port (100, 110) into two waves phase-shifted by 90° via the coupling portion 12. The two input ports of the septum polarizer 2 are thus each excited simultaneously by one of these two waves phase-shifted by 90°, thus creating two circular polarizations phase-shifted by 90°.

[0048] The two circular polarizations are then combined in the septum polarizer 2, creating in the output port 201 of the septum polarizer, a linearly polarized wave inclined at 45° relative to the input wave of the hybrid coupler 1.

[0049] Still according to this first embodiment, the first and second waveguides (10, 11) of the hybrid coupler 1 may have a rectangular section and are arranged parallel to each other, as illustrated in Figure 3. Advantageously, one of the long sides of the rectangular section of the first waveguide is parallel to one of the long sides of the rectangular section of the second waveguide, so that the largest walls rectangular side plates of the first and second waveguides are arranged in parallel planes.

[0050] The two waveguides (10, 11) are connected to each other by a coupling portion 12. More precisely, this coupling portion 12 connects one of the small rectangular side walls of the first waveguide 10 with one of the small rectangular side walls of the second waveguide 11. The two small rectangular walls are arranged in the same plane, this plane being perpendicular to the larger rectangular side walls of each waveguide. Each of the smaller walls is provided with an opening at the coupling portion so as to allow a wave to pass from the first waveguide to the second or from the second to the first. The coupling portion comprises a waveguide portion extending between these two openings so as to propagate the wave between these two openings.

[0051] In a particular embodiment illustrated in Figure 3, the waveguide portion of the coupling portion 12 has a trapezoidal geometry. More specifically, this waveguide portion comprises a rectangular base contiguous to the smaller side walls of the first and second waveguides (10, 11). The waveguide portion extends in a direction perpendicular to the rectangular base and its section parallel to the base decreases until it forms a rectangular upper face opposite the base, this rectangular face thus having dimensions smaller than those of the base.

[0052] The coupling portion 12 may be provided with one or more impedance matching elements such as grooves, internal protrusions or openings in a wall of the coupling portion. These elements are intended to optimize the transmission of signals in the coupling portion. As illustrated in FIG. 3, an opening on the upper face of the coupling portion may be provided in order to improve the transmission of a signal between the first and second waveguides (10, 11).

[0053] Figures 4a and 4b illustrate an orthomode transducer according to the first embodiment mentioned above. The hybrid coupler 1 and the septum polarizer 2 are made in one piece by additive manufacturing so that no assembly is necessary to obtain the transducer of the present invention.

[0054] In one embodiment, the printing direction of the layers for additive manufacturing coincides with the direction of propagation of the waves in the orthomode transducer. In order to reduce, or even eliminate, the need for printing supports during manufacturing, certain portions of the transducer are adapted for additive manufacturing. In particular, certain cantilevered portions are inclined so as to form an angle significantly less than 90° with the printing direction.

[0055] As illustrated in Figure 3, the coupling portion 12 of the hybrid coupler may comprise side faces forming an angle of between 35° and 55° with the direction of propagation so as to eliminate the need for any printing supports.

[0056] According to a second main embodiment, the orthomode transducer comprises a hybrid coupler 1 producing a phase shift of 180° between the two waves at the outputs of the coupler when the device operates in transmission.

[0057] According to this second main embodiment, the coupling portion 12 of the hybrid coupler 1 connects a first lateral face of the first waveguide 10 and a second lateral face of the second waveguide 11, the first and second lateral faces being arranged in separate and parallel planes.

[0058] In transmission, this hybrid coupler 1 divides a wave propagated by the first or second input port (100, 110) into two waves phase-shifted by 180° via the coupling portion 12. The two input ports of the septum polarizer 2 are thus each excited simultaneously by one of these two waves phase-shifted by 180°, thus creating two circular polarizations phase-shifted by 180°.

[0059] In an embodiment illustrated in Figures 5a and 5b, the coupling portion 12 comprises a plurality of branches 121, each branch being connected on the one hand to the first lateral face of the first waveguide 10 and on the other hand to the second lateral face of the second waveguide 11. Thus, each branch 121 intersects the plane of symmetry of the first and second waveguides (10, 11).

[0060] Each branch 121 comprises a waveguide for propagating a wave via an opening in the first waveguide 10 towards the second waveguide 11 via an opening in the wall of the second side face, or vice versa.

[0061] In an embodiment not illustrated, each branch 121 comprises a waveguide extending perpendicular to the lateral faces of the first and second waveguides (10, 11). The section of this waveguide may be triangular, square, rectangular, pentagonal, hexagonal, or more generally polygonal. The section of this waveguide may also comprise curved portions in addition to or replacing rectilinear portions.

[0062] As before, in order to reduce the use of printing media, certain portions of the hybrid coupler according to the second main embodiment are inclined with respect to the printing direction. The printing direction is illustrated in Figures 5a and 5b by the z axis and corresponds to the direction of propagation of a signal in the coupler.

[0063] In particular, the branches 121 may comprise inclined portions so as to limit the overhanging sections and thus facilitate, or even make possible, the additive manufacturing of the device.

[0064] In an embodiment illustrated in Figures 5a and 5b, each branch 121 advantageously forms a two-sided roof, each of the sides being adjacent to one or the other of the first and second waveguides (10, 11). The edge of the roof formed by the junction of the two sides is typically contained in a plane perpendicular to the printing direction z.

[0065] The inclination of the two sides may be such that the edge of the roof points towards the septum polarizer 2 or towards the first and second input ports (100, 110). In other words, the two-sided roof has a V-shaped profile pointing in one direction or the other along the printing axis z. Such a geometry of the branches 121 advantageously makes it possible to reduce the overhanging portions and therefore to facilitate their additive manufacturing, in particular by eliminating the need for printing support.

[0066] Each panel forms an angle with the lateral face of the waveguide to which it is adjacent of between 35° and 55°, preferably between 40° and 50°.

[0067] In one embodiment, the branches 121 are symmetrical with respect to the plane of symmetry of the first and second waveguides (10, 11). In other words, each side is symmetrical to the other with respect to the plane of symmetry of the first and second waveguides.

[0068] In an embodiment illustrated in Figures 6a and 6b, each branch 121 advantageously forms a double roof with two sides. More specifically, each branch 121 comprises a first roof with two sides, each of whose sides is adjacent to the first or second waveguide (10, 11) and a second roof with two sides, each of whose sides is adjacent to the first or second waveguide (10, 11). The first and second roof with two sides are connected to each other such that the edge of the first roof and the edge of the second roof are contained in the same plane. The edge of the first roof forms an angle with the edge of the second roof at the location of the connection between the two roofs.

[0069] In other words, the embodiment illustrated in Figures 6a and 6b is similar to that illustrated in Figures 5a and 5b, except that each branch of the coupling portion forms a bend in the plane containing the edges of the branches.

[0070] Each of the two edges forms an angle with the printing direction of between 35° and 55°, so that the angle between the two edges is between 70° and 110°.

[0071] In one embodiment, each branch 121 has a double symmetry. In fact, each branch has a first symmetry with respect to the plane of symmetry of the first and second waveguides (10, 11) and a second symmetry with respect to the plane perpendicular to the plane of symmetry of the first and second waveguides containing the printing direction z.

[0072] As illustrated in Figures 7a and 7b, the orthomode transducer including the branch coupler has overall symmetry about a plane containing the septum 22. Reference numbers used in the figures 1 Hybrid coupler 10 First waveguide 11 Second waveguide 100 First port of entry 101 First exit port 110 Second input port 111 Second output port 12 Coupling portion 121 Branch 122 Pan 123 Edge 2 Septum polarizer 200 Third input port 210 Fourth input port 201 Third output port 22 Septum z Print direction

Claims

Claims 1. A broadband linearly polarized orthomode transducer obtained by additive manufacturing comprising: a hybrid coupler (1) comprising: a first waveguide (10) comprising a first input port (100) and a first output port (101) with single polarization; and a second waveguide (11), comprising a second input port (110) and a second output port (111) with single polarization; and a coupling portion (12) connecting the first waveguide (10) to the second waveguide (11); the orthomode transducer further comprising: a septum polarizer (2) comprising: a third input port (200) with single polarization connected to the first output port (101); and a fourth input port (210) with single polarization connected to the second output port (111); and a third output port (201) with single polarization;wherein the first and second waveguides (10,11) are arranged symmetrically with respect to a plane of symmetry containing a septum (22) of the septum polarizer (2).; 2. Orthomode transducer according to claim 1, wherein the coupling portion (12) connects a first lateral face of the first waveguide (10) and a second lateral face of the second waveguide (11), the first and second lateral faces being arranged in the same plane.

3. Orthomode transducer according to the preceding claim, in which the first lateral face corresponds to a face of the first waveguide (10) of minimum dimension, and the second lateral face corresponds to a face of the second waveguide (11) of minimum dimension.

4. Orthomode transducer according to the preceding claim, in which the coupling portion (12) comprises a prism with a trapezoidal base, a first portion of a rectangular face of the prism being in contact with the first lateral face of the first waveguide and a second portion of the rectangular face of the prism being in contact with the second lateral face of the second waveguide.

5. Orthomode transducer according to the preceding claim, the coupling portion (12) comprising an impedance matching element.

6. Orthomode transducer according to claim 1, wherein the coupling portion (12) connects a first lateral face of the first waveguide (10) and a second lateral face of the second waveguide (11), the first and second lateral faces being arranged in separate and parallel planes.

7. An orthomode transducer according to claim 6, wherein the coupling portion (12) comprises a plurality of branches (121), a first end of each branch being connected to the first waveguide (10) and a second end of each branch being connected to the second waveguide (11).

8. Orthomode transducer according to claim 7, in which each branch (121) forms a two-sided roof, a first side of the roof being adjacent to the first lateral face and a second side of the roof being adjacent to the second lateral face.

9. Orthomode transducer according to claim 8, in which an edge of the roof formed by the junction of the first and second panes is contained in a plane perpendicular to a printing direction (z).

10. Orthomode transducer according to one of claims 8 to 9, in which the first face forms an angle with the first lateral face of the first waveguide of between 35° and 55° and in which the second face forms an angle with the second lateral face of the second waveguide of between 35° and 55°.

11. Orthomode transducer according to claim 7, in which each branch (121) forms a double roof with two sides comprising a first roof with two sides of which a first side is adjacent to the first lateral face and a second side is adjacent to the second lateral face, and comprising a second roof with two sides, of which a third side is adjacent to the first lateral face and a fourth side is adjacent to the second lateral face.

12. Orthomode transducer according to claim 11, in which the first pan and the third pan form an angle with the first lateral face included between 35° and 55° and in which the second and fourth sides form an angle with the second lateral face of between 35° and 55°.

13. Orthomode transducer according to one of claims 11 to 12, in which an edge of the first roof formed by the junction of the first and second pans forms an angle with a printing direction (z) of between 35° and 55° and in which an edge of the second roof formed by the junction of the third pan and the fourth pan forms an angle with the printing direction (z) of between 35° and 55°.