Lucille Kuhler
email: lucille.kuhler@enac.fr

Nathalie Raveu
email: raveu@laplace.univ-tlse.fr

Gwenn Le Fur
email: gwenn.lefur@cnes.fr

Luc Duchesne
email: luc.duchesne@mvg-world.com

The Modal

come

INTRODUCTION

The advantages of using metamaterials in industry are well known [START_REF] Wu | Broadband Metamaterial-Enabled Hybrid-Mode Horn Antennas[END_REF]- [START_REF] Lier | Statuts on Meta-Horn Development -Theory and Experiments[END_REF] : controlling the field distribution [START_REF] Shaw | Profiled Hard Metamaterial Horns for Multibeam Reflectors[END_REF], [START_REF] Kildal | Hard Horns Improve Cluster Feeds of Satellite Antennas[END_REF]- [START_REF] Lier | Review of Soft and Hard Horn Antennas, Including Metamaterial-Based Hybrid-Mode Horns[END_REF], reducing the size of the devices [START_REF] Pollock | Below-Cutoff Propagation in Metamaterial-Lined Circular Waveguides[END_REF]- [START_REF] Pollock | Experimental Verification of Below-Cutoff Propagation in Miniaturized Circular Waveguides Using Anisotropic ENNZ Metamaterial Liners[END_REF], controlling the polarization [START_REF] Ma | Single-layer circular polarizer using metamaterial and its application in antenna[END_REF]- [START_REF] Zhu | Design of polarization reconfigurable antenna using métasurfaces[END_REF] and maybe in the next years for cloaking [START_REF] Schurig | Metamaterial Electromagnetic Cloak at Microwave Frequencies[END_REF] or reducing the antennas radar cross-section [START_REF] Jia | Broadband polarization rotation reflective surfaces and their applications to RCS reduction[END_REF]- [START_REF] Sharma | Design of a wideband polarisation conversion metasurface and its application for RCS reduction and gain enhancement of a circularly polarised antenna[END_REF]. With their structuration, they constitute new artificial material with electromagnetic properties that are not available in nature [START_REF] Shelby | Experimental Verification of a Negative Index of Refraction[END_REF], [START_REF] Smith | Composite Medium with Simultaneoustly Negative Permeability and Permittivity[END_REF]. In [START_REF] Smith | Composite Medium with Simultaneoustly Negative Permeability and Permittivity[END_REF]- [START_REF] Ziolkowski | Wave Propagation in Media Having Negative Permittivity and Permeability[END_REF], a relative permittivity and/or a permeability lower than 1 or less than 0 have been achieved. Different methods allow the characterization of metamaterials: by approximation of their equivalent relative permittivity and permeability, or with their surface impedances at any given height [START_REF] Wu | Nature-Inspired Design of Soft, Hard and Hybrid Metasurfaces[END_REF]- [START_REF] Warecka | Efficient Finite Element Analysis of Axially Symmetrical Waveguides and Waveguide Discontinuities[END_REF]. In the space industry, the main advantage of using metamaterials lies in the reduction of the size and the weight of horn antennas and waveguides [START_REF] Shaw | Profiled Hard Metamaterial Horns for Multibeam Reflectors[END_REF], [START_REF] Kildal | Hard Horns Improve Cluster Feeds of Satellite Antennas[END_REF], [START_REF] Thomas | A Method of Synthesizing Radiation Patterns with Axial Symmetry[END_REF], [START_REF] Pollock | Below-Cutoff Propagation in Metamaterial-Lined Circular Waveguides[END_REF]- [START_REF] Pollock | Experimental Verification of Below-Cutoff Propagation in Miniaturized Circular Waveguides Using Anisotropic ENNZ Metamaterial Liners[END_REF]. In [START_REF] Byrne | Reduction of Rectangular Waveguide Cross-Section with Metamaterials: A New Approach[END_REF], the rectangular waveguide cross section has been reduced using a new method: the Modal Expansion Theory (MET).

This method has been developed over the past few years, [START_REF] Byrne | Etude et Conception de Guides d'Onde et d'Antennes Cornets à Métamatériaux[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF], [START_REF] Byrne | Dispersion Properties of Corrugated Waveguides Based on the Modal Theory[END_REF], [START_REF] Raveu | Modal Theory for Waveguides with Anisotropic Surface Impedance Boundaries[END_REF]. With the MET, the propagation characteristics of cylindrical and rectangular waveguides with anisotropic walls are quickly obtained. Moreover, with a combination of the MET code and a 2-D Finite-Element-Method (FEM) code, it is now possible to deal with waveguides with 2-D metamaterial walls, [START_REF] Byrne | Modal analysis of Rectangular Waveguides with 2D Metamaterials[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF]. The 2-D FEM code computes the equivalent surface impedances of the metamaterial. These impedances are dependent on the frequency, incidence angle and propagation mode. In these articles [START_REF] Byrne | Modal analysis of Rectangular Waveguides with 2D Metamaterials[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF], the MET accuracy and the time efficiency have been well demonstrated, up to 360 time faster than the commercial software HFSS. However, with this 2-D FEM code, only 𝑚 = 0 order modes can be determined. Even if such modes are sometimes used in the space industry -for example, the TM 01 mode with a circular symmetric pattern that is used in signals detection [START_REF] Verma | Design of a Shaped Omni Directional Circular Waveguide Antenna[END_REF], or in communications and satellites positioning [START_REF] Tang | A Low Sidelobe and High Gain Omni-Directional COCO Antenna Array[END_REF], [START_REF] Güngör | Design of a Verticaly Polarized Omni-Directional Antenna at Ka-Band[END_REF] -the fundamental mode is often the TE 11 and no hybrid mode can be addressed. Hybrid modes are often wanted as fundamental mode. As a matter of fact, their radiation pattern is axially symmetric with zero cross-polarization response and may be shaped for high efficiency [START_REF] Thomas | A Method of Synthesizing Radiation Patterns with Axial Symmetry[END_REF]. Horn antennas may support such modes [START_REF] Granet | Design of Corrugated Horns: a Primer[END_REF]- [START_REF] Clarricoats | Corrugated Horns for Microwave Antennas[END_REF]. In [START_REF] Warecka | Efficient Finite Element Analysis of Axially Symmetrical Waveguides and Waveguide Discontinuities[END_REF], the authors propose a 2-D FEM code returning the whole dispersion diagram of cylindrical waveguides with 2-D metamaterials compared to [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF]. However, this code [START_REF] Warecka | Efficient Finite Element Analysis of Axially Symmetrical Waveguides and Waveguide Discontinuities[END_REF] and the 2-D MET [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF] are not adequate to characterize all metamaterials. For this reason and to take all order modes into account the MET has been hybridized with a 3-D FEM code. Thus, waveguides with all metamaterials (with or without 𝜃invariance) could be characterized through the MET.

In this paper, a new 3-D hybrid numerical technique is proposed to achieve these goals for cylindrical waveguides. The MET is used to obtain the propagation characteristics of metamaterial waveguides with a new 3-D FEM solution proposed to compute the surface impedances. This technique is carefully explained in Section II. In the final section, the validation of this method is presented. The comparison point is performed with the FEM commercial software HFSS. Three different cylindrical waveguides are tested: a transversal corrugated waveguide studied in [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF], [START_REF] Byrne | Dispersion Properties of Corrugated Waveguides Based on the Modal Theory[END_REF], a longitudinal corrugated waveguide and the finally a peak-structure metamaterial waveguide.

HYBRID NUMERICAL TECHNIQUE FOR 3D-METAMATERIAL WAVEGUIDES CHARACTERIZATION

In this article, the studied waveguides are cylindrical waveguides with metamaterial walls and an invariance along the 𝑧-axis, see Fig. 1. Consequently, the electromagnetic field has an 𝑒 !! ! ! dependence, where 𝛾 ! is the propagation constant along the 𝑧-axis.

To characterize this kind of waveguides the MET is used [START_REF] Byrne | Modal analysis of Rectangular Waveguides with 2D Metamaterials[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF]. Indeed, the metamaterial structuration -whose period is supposed small compared to the wavelength [START_REF] Smith | Composite Medium with Simultaneoustly Negative Permeability and Permittivity[END_REF] -enables their study by computing equivalent anisotropic surface impedances. Therefore, the hybrid technique proposed, in this paper, is composed of the MET main program that solves the dispersion equation and a program in the open-source software GetDP [START_REF] Dular | GetDP Reference Manual: The Documentation for GetDP 3.0 A General Environment for the Treatment of Discrte Problems[END_REF] that computes the surface impedances.

The Modal Expansion Theory

In the cylindrical coordinate system, the 𝑍 ! ! and 𝑍 ! ! surface impedances are defined by [START_REF] Wu | Broadband Metamaterial-Enabled Hybrid-Mode Horn Antennas[END_REF].

𝑍 ! ! = - = 0, (2)
where 𝑎 is the waveguide internal radius, 𝑢 ! = 𝑘 ! 𝑎, 𝑍 ! the free space characteristic impedance, 𝐽 ! the Bessel function of order 𝑚, and 𝐽 ! ! the derivative of the Bessel function 𝐽 ! . In [START_REF] Byrne | Etude et Conception de Guides d'Onde et d'Antennes Cornets à Métamatériaux[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF], [START_REF] Raveu | Modal Theory for Waveguides with Anisotropic Surface Impedance Boundaries[END_REF], it was pointed out that the 𝑍 ! ! and 𝑍 ! ! surface impedances were dependent on the 𝜑 incidence angle, defined with the relation [START_REF] Shaw | Profiled Hard Metamaterial Horns for Multibeam Reflectors[END_REF].

𝜑 = arcsin 𝛽 𝑘 ! , (3)
where 𝛽 = -𝑗𝛾 ! , since only propagating modes were taken into account. In [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF], the MET with the 2-D FEM code has been extended to identify evanescent modes. Further research will be lead to characterize these modes with the 3-D FEM hybridized MET. Consequently, the MET algorithm has to solve the dispersion equation with the recursive solution proposed in the Fig. 2. In the algorithm third step and for each 𝜑 ! computed angle, the 𝑍 ! ! and 𝑍 ! ! surface impedances are computed thanks to the GetDP call.

The 3D-Conformal Metamaterial Modelling

The analyzed unit cell is simplified in GetDP compared to HFSS solution since it allows non parallel periodic walls definition. This new 3-D unit cell is introduced in the next section. Then the code and its implementation in the MET are explained.

The 3D Unit Cell

The waveguides under consideration are, obviously, periodical along the 𝑧-axis. In this paper, the metamaterial is not 𝜃-invariant anymore, compared to [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF]- [START_REF] Kuhler | Théorie Modale Élargie Appliquée aux Guides d'Onde Cylindriques à Métamatériaux[END_REF], but periodically set along this axis. The Fig. 3a) represents a 3-D metamaterial waveguide, with two planes that defines the angular periodicity of the metamaterial. Thus one periodic lattice can be isolated between these planes.

With the Fig. 3a) example, the 3-D metamaterial appears 18 times at a given longitunal position, thus the elementary lattice can be defined with an 𝛼 angle of 20°. The 3-D unit cell is represented on the Fig. 3b).

The lattice is isolated along the 𝑧-axis and 𝜃-axis. A periodical condition ( 4) is imposed between the Γ ! and Γ ! walls. A similar condition [START_REF] Kildal | Hard Horns Improve Cluster Feeds of Satellite Antennas[END_REF] is introduced between the Γ ! and Γ ! walls.

𝑈 𝜌, 𝜃, 𝑧 + 𝑝 ! ! = 𝑈 𝜌, 𝜃, 𝑧 ! ! × exp -𝛾 ! 𝑝 , (1)
𝑈 𝜌, 𝜃 + 𝛼, 𝑧

! ! = 𝑈 𝜌, 𝜃, 𝑧 ! ! × exp -𝑗𝑚𝛼 , (2)
where 𝑈 is a vector which could be the magnetic field 𝐻 or the electric field 𝐸, 𝑝 the distance between Γ ! and Γ ! , 𝛼 the angle between Γ ! and Γ ! (as defined on the Fig. 3b), and 𝑚 the mode order. A surface impedance can be defined on Γ ! . In our cases, a PEC condition (the surface impedance is equal to 0) is applied to this boundary. The 𝑍 ! ! and 𝑍 ! ! (1) surface impedances are computed on the Ω ! plane. ℎ is the height to the Ω ! plane (ℎ = 𝐴 -𝑎).

The New 3D-Finite Element Method Code

GetDP [START_REF] Dular | GetDP Reference Manual: The Documentation for GetDP 3.0 A General Environment for the Treatment of Discrte Problems[END_REF] means General environment for the treatment of Discrete Problems. This program is used to compute the equivalent surface impedances of the metamaterial. In its main file, the structure definition and the problem equations are specified.

The periodical conditions (4) between Γ ! and Γ ! and ( 5) between Γ ! and Γ ! are set in the code as the PEC condition on Γ ! . The problem is solved by using a FEM formulation with edge elements. The weak formulation (6) used is the following:

∇ ! 𝐸 + 𝑘 ! ! 𝐸 𝑣𝑑𝑉 = 0 ! , (3)
where 𝑣 is the test function, since the Galerkin method is used, and 𝑉 the volume described by the unit cell. Thanks to a post processing operation, the electric and magnetic fields are returned in the whole structure. Besides a file is created with the values of these fields on the Ω ! plane. This file is used in the MET algorithm to compute the surface impedances.

Implementation of the 3D-Code in the MET

As explained before, the GetDP program is called on the third step in the MET algorithm. Hence, calling it each iteration can be a disadvantage, as it will take a significant computation time. Nevertheless, in the next part, it will be proven that even using this scheduling the time efficiency is better than using a commercial software.

RESULTS

To validate this method, this hybrid numerical technique is firstly applied to a waveguide with a 2-D metamaterial. This waveguide has corrugations along the 𝜃-axis [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF], [START_REF] Byrne | Dispersion Properties of Corrugated Waveguides Based on the Modal Theory[END_REF]. Subsequently, a 3-D metamaterial is studied: corrugation along the 𝑧-axis. Finally, the code is applied to a waveguide with a peak-structure metamaterial. All the dispersion diagrams are compared to those obtained with HFSS. All simulations were made with the same computer [Intel® Core™ i7-7700 CPU @ 3.60 GHz, 16 GB of RAM].

HFSS validation tool

The eigenmode solver is chosen for this problem. A 3-D unit-cell, see Fig. 4, is created and periodic boundary conditions are added. A phase delay is included between these two walls, with a scan from 0° to 180°. This solver returns the resonance frequency of each solution (corresponding to a phase delay) and each mode. Then the propagation constant is computed using the phase delay and the 𝑝 distant between both walls. The analysis setup of the simulations changes with the tested waveguide.

Waveguide with corrugations along the 𝜽-axis

The studied waveguide is represented on the Fig. 5a). The Figure 5b) shows the corrugation dimensions and the mesh obtained with GMSH-GetDP [START_REF] Dular | GetDP Reference Manual: The Documentation for GetDP 3.0 A General Environment for the Treatment of Discrte Problems[END_REF], [START_REF] Geuzaine | Gmsh: A three-dimensional finite element mesh generator with built-in preand post-processing facilities[END_REF]. In [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF], only 𝑚 = 0 order modes were displayed on the dispersion diagrams, and in [START_REF] Byrne | Dispersion Properties of Corrugated Waveguides Based on the Modal Theory[END_REF] all modes were found, however the results between the method proposed and HFSS did not perfectly coincide. The Figure 6 represents the new dispersion diagrams obtained with the MET and the GetDP code (dots).

Both diagrams coincide and all modes are found. Consequently, the MET with the hybrid numerical technique is validated on this example. In HFSS, the adaptive solution for this waveguide is set to a maximum Δ𝑓 of 0.01%. Hence, with HFSS the simulation lasts three days, while with the MET it lasts 4 hours and 30 min. As explained before, the computation time with the MET with GetDP is quite longer than the MET with the 2-D FEM code [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF] (10-min computation time), represented with triangles on the Fig. 6, due to the fact the software is called each iteration. Moreover, the frequency step is not the same. Indeed, in the MET the frequency is imposed then the incidence angle is evaluated to give the propagation constant. In HFSS, the incidence angle is imposed and the frequency evaluated. Figure 6. Dispersion diagrams of the cylindrical waveguide with the corrugation presented in Fig. 5a) obtained with the MET + 3-D FEM code (dots), the MET + 2-D FEM code [START_REF] Kuhler | The Propagation Characteristics of 2-D Metamaterial Waveguides Using the Modal Expansion Theory[END_REF] (triangles) and HFSS (circles).

Waveguide with corrugations along the 𝒛-axis

Corrugations along the 𝑧-axis are now inserted in a waveguide, see Fig. 7a). The 𝛼 angle also conditions the number of metamaterial. In this example, 𝛼 = 30°, thus, there are twelve corrugations (360°/𝛼 = 12). The different dimensions and the mesh of the 3-D unit cell used in the MET are represented on the Fig. 7b).

The algorithm is applied to this waveguide, and the dispersion diagrams are compared to HFSS ones on the Fig. 8. It took 40 min to obtain the dispersion diagrams with HFSS while with the MET only required a 21-min computation time. Therefore, the MET is still 2 time faster. It can be noted that the simulation on HFSS is drastically reduced compared to the previous case. As the matter of fact, for this waveguide Δ𝑓 !"# = 0.05%, otherwise the simulation crashed due to a lack of memory space. The difference in the dispersion diagrams between both methods is small: the gap between the MET and HFSS increases with the 𝑚 order mode.

Waveguide with peak-structure metamaterial

Finally, a 3-D peak-structure metamaterial waveguide is tested with this hybrid numerical technique, see Fig. 9a). The Fig. 9b) (i) represents the dimensions and the mesh used in the MET and the Fig. 9b) (ii) displays the PEC part of the unit cell. As 𝛼 = 20°, there are 18 peakstructures in a cross-section. This new method has allowed plotting the whole dispersion diagrams of all kind of metamaterial waveguides. Moreover, the proposed method is at least twice as fast as the FEM commercial software for 0.05% error criteria on HFSS and 16 times faster for 0.01% error criteria.

CONCLUSION

In this article, the MET has been uploaded with a hybrid technique using an open software: GetDP. The new algorithm has been successfully applied to three different waveguides: one with a 2-D metamaterial (a corrugation along the 𝜃-axis), and two with a 3-D metamaterial. As a consequence, the MET is now completely developed to characterize cylindrical waveguides with various metamaterials. Furthermore, the computation time is significantly reduced compared to commercial software.

The MET is currently used to reduce the cross-section of a cylindrical sensor working at the frequency band 6, 8 GHz. As a matter of fact, the reduction of the cross-section of sensor is mandatory to decrease its impact on the measurement of a testing antenna.

Moreover, the characterization of evanescent modes is also under development. Since, to characterize horn antenna with a mode-matching technique, these modes are also required. With the mode-matching technique, it will also be possible to deal with metamaterials that changes along the 𝑧-axis. Such kind of metamaterial are useful to improve the matching between the feeding access and the waveguide, or to avoid an abrupt termination of a horn antenna or an open-ended waveguide.

Figure 1 .

 1 Figure 1. Cylindrical waveguide with anisotropic walls.

Figure 2 .

 2 Figure 2. Schematic algorithm to correct the 𝜑 incidence angle.

Figure 3 .

 3 Figure 3. a) 3-D metamaterial waveguide and b) the 3-D unit cell.

Figure 4 .

 4 Figure 4. Cylindrical representation of a waveguide with periodic boundary conditions and anisotropic surface simulated in HFSS.

Figure 5 .

 5 Figure 5. a) Corrugated waveguide. The 3-D blue dashed section is used in HFSS, and the 3-D red part is used in the MET b) represented with the mesh. The dimensions are 𝐴 = 100 mm, 𝑎 = 80 mm, ℎ = 20 mm, 𝑝 = 26.225 mm, 𝑤 = 20.98 mm, 𝑑 = 18.2 mm and 𝛼 = 20°.

Figure 7 .

 7 Figure 7. a) Corrugated waveguide along the 𝑧-axis. The 3-D blue dashed section is used in HFSS, and the 3-D red part is used in the MET b) represented with the mesh. The dimensions are 𝐴 = 100 mm, 𝑎 = 80 mm, ℎ = 20 mm, 𝑝 = 26.225 mm, 𝑑 = 18.2 mm and 𝛼 = 30°.

Figure 8 .

 8 Figure 8. Dispersion diagrams of the cylindrical waveguide with the corrugation presented in Fig. 7a) obtained with MET (dots) and HFSS (circles).

Figure 9 .

 9 Figure 9. a) Waveguide with the peak-structure metamaterial. The 3-D blue dashed section is used in HFSS, and the 3-D red part is used in the MET b) represented with the mesh. The dimensions are (i) 𝐴 = 30 mm, 𝑎 = 15 mm, ℎ = 15 mm, 𝑝 = 10 mm; (ii) for the Γ ! part (the PEC boundary) of the 3-D unit cell with 𝑑 = 8 mm, 𝛼 = 20°, and 𝑤 = 8 mm.

Figure 10 .

 10 Figure 10. Dispersion diagrams of the cylindrical waveguide with the corrugation presented in Fig. 9a) obtained with MET (dots) and HFSS (circles).

ACKNOWLEDGMENT

This work was made in collaboration between the CNES, MVG Industries, and the LAPLACE Laboratory. Moreover, the authors would like to thank Pablo Stuardo and Miguel Rojas Sanchez for their precious help in the development of the GetDP program.