A Fast and Accurate Method of Synthesizing X-Wave Launchers by Metallic Horns

This paper presents a method for synthesizing metallic horns capable of radiating highly localized electromagnetic pulses, known as X-waves. The proposed method is based on a mode-matching framework for shaped horns. The aperture field distribution required to generate X-waves is synthesized by mode conversion. The near-field horns generated by this method are full-metal structures radiating X-waves in a wide range of frequencies with low dispersion. The obtained structures may be scaled to any frequency range, and they are particularly suited to millimeter and sub-millimeter wave applications. We validate the concept by presenting a horn capable of radiating X-waves over a 44% fractional bandwidth (FBW) with a 5° dispersion of the axicon angle. Capabilities and limitations of this design procedure and the synthesized launchers are demonstrated, discussed, and compared with other state-of-the-art solutions.


I. INTRODUCTION
In past decades, Bessel beams have received large attention by the physics and engineering communities. Among their applications in optics and in microwave engineering, one finds medical imaging [1], wireless power transfer [2], and electron microscopy [3], to name a few. Ideal Bessel beams, which were introduced in 1987 by Durnin [4], are exact solutions of the Helmholtz equation that do not suffer from diffractive spreading while propagating. They are defined as follows (ρ, φ, z, t) = J n (χρ)e inφ e i(k z z−ωt) , where denotes a solution of the scalar Helmholtz wave equation, ω is the angular frequency, and (χ, k z ) are the radial and longitudinal components of the wave vector k, respectively, such that k 2 = χ 2 + k 2 z = (ω/c) 2 . In turn, J n (χρ) stands for the standard Bessel function of the first kind and n-th order, with n an integer number. The variables ρ, φ and z are the cylindrical coordinates, with z being the assumed direction of propagation. Theoretically, ideal Bessel beams The associate editor coordinating the review of this manuscript and approving it for publication was Vittorio Degli-Esposti . require infinite energy for their realization, namely, infinite radiating apertures. However, finite apertures can generate truncated Bessel beams which resist diffraction effects over a limited distance known as the ''non-diffractive'' range, limited by the axicon angle θ =arctan(χ /k z ).
The polychromatic superposition of Bessel beams creates X-waves, which consist of electromagnetic pulses that resist diffraction and remain confined in both the transversal and longitudinal directions as they propagate up to a certain distance, after which they experience diffraction. Ideal X-waves are defined by the expression [17] χ(ρ, z, t) = include secure short-range wireless communication [18] and biomedical ultrasound imaging [19].
Just like their monochromatic Bessel counterpart, the synthesis of ideal X-waves requires infinite apertures. In other terms, finite apertures can radiate truncated X-waves over limited frequency bands. We will refer to the finite frequency band in which truncated X-waves are generated as the operating frequency band, and to its ratio to the central frequency of the band as the fractional bandwidth (FBW). Unlike for Bessel beams, there exist only a few experimental demonstrations of X-wave generation in literature. This is due to inherent difficulties in realizing Bessel beam launchers with low-dispersive axicon angles θ over a wide frequency band. The first realization of X-waves was done in acoustics [20], followed by other proofs of concept in optics [21], [22].
The first experimental demonstration of X-waves in the microwave frequency range (from 19 to 29 GHz) was reported in 2018 [23]. The presented launcher consists of a CNC-milled dielectric lens surrounded by a copper tape fed by a coaxial cable. The launcher is capable of radiating X-waves over a 42% FBW with a 2 • dispersion of the axicon angle θ. Despite these excellent characteristics, the use of dielectrics in [23] renders this realization lossy at higher frequencies. A similar demonstration of X-waves has been reported in [24], where the launcher uses a 3D printed dielectric metamaterial instead of the dielectric lens. This prototype can radiate X-waves over a 50% FBW with a 7 • dispersion of the axicon angle θ. As in [23], the use of dielectrics can make such a solution lossy in the millimeter and sub-millimeter wave regimes. The use of a radial parallel-plate waveguide antenna was also proposed for generating X-waves and experimentally validated in [25]. The main drawback of [25] is the dispersive nature of radial launchers (24 • for a 22% FBW). Capabilities of RLSA launchers based on Bessel and Hankel aperture distribution to radiate X-waves were theoretically analyzed in [26]. The drawback of [26] is, as in [25], the high dispersion of the axicon angle in RLSA launchers.
In this paper, we present a solution to generate X-waves by metallic horns. Metallic horns are a very common solution for transmitting radio waves due to their wide bandwidth, radiation characteristics, and simple construction and excitation. An overview of metallic horns and their applications, from their historical beginnings to current research activities, can be found in [27]. In previous decades, many techniques have been proposed to improve the performance of metallic horns. One of the most common approaches includes modifying the metallic profile of the horn by introducing corrugations [28], [29]. Some excellent characteristics of corrugated horns have been reported, such as dual-band operation [30], high Gaussicity and a very low sidelobe level (up to -60 dB) [31], large bandwidth [32], and dual-polarization [33].
To avoid the manufacturing issues of corrugated horns, Granet et al. proposed in 2004 smooth-walled spline-profiled metallic horns [34]. Such horns are designed by optimizing their profile to match the far-field to a specified radiation pattern. Excellent characteristics of spline-profiled horns have also been reported, such as low cross-polarization [35], multiband operation [36], low sidelobe level [37], and high Gaussicity at THz frequencies [38], [39]. Performance of such antennas can be further improved by, for instance, integrating a dielectric lens [40], [41].
In this contribution, we optimize the shape of a smooth profile horn to generate X-waves in the near field. More precisely, the goal is to shape the profile to convert an axisymmetric mode at the input into an axisymmetric mode at the aperture capable of radiating X-waves. The design procedure is based on an ad-hoc mode-matching framework. Such horns are broadband, low-dispersive, do not require dielectrics or focusing elements to generate X-waves, and are scalable to any frequency range, making them particularly suited to millimeter and sub-millimeter wave applications [42] The paper is organized as follows: Sections II and III present, respectively, the numerical formulation and the practical implementation of the proposed technique. Section IV presents an example of a transvere electric (TE) X-wave launcher with 44% FBW and 5 • dispersion of the axicon angle generated by the proposed method. The capabilities and limitations of the presented procedure and related launchers are discussed in Section V and compared to other stateof-the-art launchers. Section VI demonstrates the capability of this technique to generate transverse magnetic (TM ) X-wave launchers. To conclude, final remarks are reported in Section VII.

II. DESIGN CRITERIA
It has been shown in [43] that TE 0n and TM 0n modal field distributions in circular waveguides correspond to azimuthally and radially polarized truncated Bessel beams. Without loss of generality, in the following sections we will focus on TE 0n modal field distributions in circular waveguides. The proposed methodology can also be applied to TM 0n field distributions, as discussed in Section VI.
The non-diffractive range of truncated Bessel beams is limited by the axicon angle θ = arctan(χ /k z ) and given by the expression where a is the radius of the aperture and k is the magnitude of the wave vector k. From (3), it is clear that larger non-diffractive ranges correspond to smaller transverse propagation constants/axicon angles and larger apertures, as skecthed in Fig. 1. Therefore, once the radiating aperture has been fixed, the problem of designing Bessel beam launchers with large non-diffractive ranges reduces to synthesizing TE 0n modes on circular apertures with smaller values of the axicon angle θ. Additionally, for TE 0n modes χ = χ 0n = x 0n /a, where x 0n is the zero of the derivative of the Bessel function of the corresponding mode. It is thus clear that lower-order modes require smaller apertures for achieving the same non-diffractive distance compared to higher-order modes. Superposing truncated Bessel beams over a limited range of frequencies leads to the generation of truncated X-waves, as long as θ is low-dispersive. In [17], the criteria for synthesizing well-confined X-waves is defined by a ''confinement factor'', given by where j 0,1 = 2.4048 identifies the first zero of the J 0 function, m = a/λ 0 is the normalized aperture radius in terms of wavelengths and ω = ω/ω 0 is the fractional operative Bessel beams bandwidth. The distance up to which such truncated X-waves resist diffraction effects corresponds to the non-diffractive range z max at the central frequency of the observed frequency band ω.
To summarize, the problem of synthesizing X-waves reduces to synthesizing TE 0n modes on circular apertures sufficiently large and over a sufficiently wide frequency range to satisfy (7) with a low-dispersive axicon angle θ in the entire frequency band. For instance, one can verify using (7) that an aperture with a radius of a = 3λ and an axicon angle of θ = 20 • requires at least 10% of fractional bandwidth. We will use this criteria as a practical guideline for synthesizing X-wave launchers, as demonstrated in the following sections.

III. DESIGN PROCEDURE
For synthesizing a field distribution capable of radiating X-waves, we have resorted to a known property: an axisymmetric TE 0n mode at the junction of two circular waveguides of different radii will excite only the other axisymmetric TE 0m modes, with n and m being arbitrary integer numbers [44]. Applying this property, the design procedure consists in creating a structure which converts a TE 01 mode at the input, which we assume we can generate correctly, to another TE 0m mode at the aperture, while satisfying the previously defined criteria for radiating X-waves. A simple way to obtain a circular TE 0m mode consists of a series of monotonically increasing constant waveguide sections, which we convert to a continuous profile.
To simplify and accelerate the design process, we developed an in-house mode-matching tool [44] for calculating the scattering matrix S, near-field, and far-field radiation of metallic horns with azimuthal symmetry. The tool approximates continuous antenna structures as a series of electrically short and concatenated cylindrical waveguides, with cylinder radii changing on each waveguide section. We choose the length of the cylindrical waveguide electrically small (typically less than λ/10) to assume that the resulting structure is continuous, as sketched in Fig. 2. Approximating the structure in such a way allows one to analyze it with a standard mode matching procedure [44]. We match the transverse field distribution at each discontinuity of the horn and cascade it section by section. This way, we can compute the aperture field distribution for a given profile, and consequentially the near and far-field distributions of the resulting structure. In this analysis, we assume that the structure includes a perfectly conducting infinite ground plane. Since the E-field vanishes on such a ground plane and is negligible beyond, we can assume that the E-field becomes zero beyond the radiating aperture. This modal approach is much faster than other full-wave methods (e. g. finite elements or finite integral technique [45]) and, despite the adopted approximations, the obtained results are in very good agreement with such methods.
In the following, we will assume that only the higher-order TE 01 mode is present in the input circular waveguide. One can guarantee this behavior using well-known mode converters, e.g., [46]- [48], to transform the fundamental mode of a rectangular waveguide to a TE 01 circular waveguide mode. The Marié converter in [46] offers excellent performance in terms of purity of the output TE 01 mode, while being well-matched over a large band of frequencies. This mode converter is made of metal, allowing flexibility in terms of the operating frequency range. Inherent drawbacks of the Marié mode converter are its electrical length, which can be a potential problem if used at lower frequencies, and its geometrical complexity. However, it has been shown that these limitations can be conveniently overcome by using additive manufacturing [48].
The targeted TE 0n field distribution can be chosen arbitrarily. As mentioned before, larger non-diffractive ranges can be achieved by using either larger apertures or by smaller transverse propagation constants. Therefore, it is generally convenient to design an electrically smaller antenna and target the field distribution of a lower-order mode (TE 01 or TE 02 ) on its aperture. In particular, we optimize the depths and radii of multiple waveguide sections to couple the aperture field distribution to the field distribution of a TE 01 or a TE 02 mode. More precisely, we require S 21 > 0.92 between the incident TE 01 mode and the targeted output TE 01 or TE 02 mode at each frequency point (85% of power coupling). Such constraint proved to be sufficient for the effective radiation of Bessel beams/X-waves. The second goal of the optimization was to match the synthesized horn to the input TE 01 mode. The targeted value of the input reflection coefficient is S 11 ≤ −15 dB. We also impose a design constraint on the Marié mode converter in the optimization routine, requiring that the output radius of the mode converter (input of the horn) guarantees the TE 02 mode in cutoff, so only the TE 01 mode can propagate. Due to the large number of optimization variables, we used the interior-point method in the optimization procedure to minimize the following cost function where S error 21 (f ) and S error 11 (f ) are distances between the achieved and targeted S parameters (L1 norm) at the frequency point f and W is the weight coefficient used to control the priority of the factors. The polynomial complexity of the interior point method makes it a more appropriate choice for this type of large-scale non-linear optimization problems in comparison with other similar methods (e. g. simplex).
The initial structure has a conical profile with prescribed length and input and output radii. The input radius is initially fixed to be smaller than r TE 02 , where r TE 02 is the smallest radius that supports a propagating TE 02 mode at the highest in-band frequency. In turn, the output radius a is calculated using (3) to provide the targeted non-diffractive range z max . The initial length of the horn is related to the targeted non-diffractive range and possible starting values are provided in Section V. Then, the conical profile is approximated as a series of monotonically increasing waveguide sections. Each waveguide section is defined by 2 optimization variables: its depth and radius. Typically, the number of used sections N should be greater than L/λ 0 , where L is the length of the horn. The optimization procedure then modifies the structure by tuning the 2N optimization variables until the value of the cost function (5) is zero. One iteration cycle of the optimization process consists in the following consecutive steps: • Generate the profile of the structure by defining depths and radii of cascading waveguide sections, • Calculate the S 11 of the structure for the TE 01 mode at the input, • Calculate the coupling between the incident TE 01 mode and the targeted output TE 01 or TE 02 modes, • Calculate the value of the cost function using (5), • If the goal value of the cost function is not satisfied (F c > 0), the profile of the structure is modified by changing the 2N variables according to the used optimization algorithm and the cycle is repeated. If the goal value of the cost function is satisfied (F c = 0), the optimization procedure stops.

IV. NUMERICAL VALIDATION
The design procedure was validated by synthesizing a TE 02 launcher capable of radiating Bessel beams over a 44% of fractional bandwidth. Fig. 3 presents the exact dimensions of the realized profile and the synthesized metal horn. Since the launchers designed by the procedure are full-metal structures, they may be scaled to any frequency range. Therefore, all results in this and the following sections will be in terms of free-space wavelengths λ 0 at the central frequency f 0 of the operating frequency range. The input radius of the launcher is r 0 = 0.91λ 0 , which satisfies the aforementioned design constraint of the Marié mode converter, requiring that the TE 02 mode is in cutoff in the entire frequency range. The radius of the aperture is a ≈ 5.1λ 0 , which corresponds to z max ≈ 23λ 0 of non-diffractive range at the central frequency and 12 • axicon angle θ at the central frequency with 5 • dispersion in the operating bandwidth. The total length of the launcher is around 12.72λ 0 , and consists of 20 waveguide sections, with depths varying between 0.36 and 2.18λ 0 , which we convert to a continuous profile shown in of Fig. 3. This number of sections proved to be sufficient in this particular case. The used optimization procedure was the one described in Section III, with total optimization time being around 10 minutes. The optimization was performed on an Intel Xeon 2.6 GHz dual-core CPU with 128 GB RAM. The confinement factor of the launcher (7) is around 0.08, allowing the radiation of well-confined pulses. The widths of the generated pulse in transversal and longitudinal directions, which correspond to the null-to-null distance of the amplitude profile over the axes, are S ρ = 3.72λ 0 and S z = 4.7λ 0 , respectively. The pulse widths were evaluated using following expressions [17] S ρ = 2j 0,1 c ω 0 sin θ ,  Finally, the S 11 of the launcher is below -20 dB in the entire frequency range. Next, we compare the performance of the X-wave launcher with full-wave simulations. First, we demonstrate the non-diffractive behavior of the structure.  [45]. The edges of the operating band correspond to frequency points in which the required coupling to the TE 02 mode is not satisfied and the resulting radiated field distributions do not resemble Bessel beams anymore. Fig. 5 presents a comparison of the E y along the x-axis at 9.2, 13.8, and 18.4λ 0 above the aperture of the launcher at the central frequency. Figs. 4 and 5 demonstrate the non-diffractive behavior of the launcher in the prescribed range of frequencies. The excellent agreement between the results obtained with our numerical tool and CST's time domain solver in Fig. 5 validates the tool. Fig. 6 presents the instantaneous intensity of the pulse (i. e. longitudinal component of the H -field), confirming the capability of the launcher to radiate well-confined X-waves and thus also validating the design procedure.

V. PERFORMANCE GUIDELINES
In order to determine the capabilities and limitations of the mode-converting procedure, we have synthesized X-wave launchers with TE 01 and TE 02 aperture field distributions.  We have considered specified values of non-diffractive range and a set of FBW for the synthesized launchers, while the input radius is fixed to support the propagation of a TE 01 mode. In particular, we synthesized launchers with non-diffractive ranges spanning from 6 to 60 λ 0 with 11, 22, 33, and 44% FBW (with the same central frequency), with a fixed input radius r 0 = 0.91λ 0 . The 44% FBW corresponds to the maximum value which we were able to achieve using the proposed optimization procedure. Since we are interested in determining the limits of this design technique, we modified the optimization routine to force the synthesized launchers to be as short as possible. We repeated the optimization routine several times for every specified value of the non-diffractive range to find the corresponding minimal length of the launcher, and the obtained values are displayed in Fig. 7.
From Fig. 7, one can note some important properties of the mode-converting procedure. Firstly, it is clear that to achieve longer non-diffractive ranges for the generated Bessel beams/X-waves, longer launchers are required. A peculiar property appears from Fig. 7: the ratio between the non-diffractive range and the minimal length of the launcher capable of radiating such a beam is almost linear. Longer launchers are also required to generate Bessel beams in a larger frequency band. Additionally, we can achieve the same non-diffractive range of the generated beam with a significantly smaller launcher by synthesizing a lower-order TE 0n mode field distribution at the aperture. Finally, we were not able to achieve sufficient coupling to the TE 02 for lower values of the non-diffractive range for 44% of FBW. This is most likely because the corresponding aperture radii become closer to the cutoff of the TE 02 mode at the lower end of the frequency band, making it difficult to achieve the sufficient coupling to the TE 02 mode without distorting field distributions at the higher end of the band. Notwithstanding this limitation, the results in Fig. 7 can be readily used to define the starting profile for the optimization in Section III. Moreover, these results demonstrate that the procedure is flexible in terms of the operating frequency band and also in terms of performances of the generated launchers. One can also note that, other than the trade-off between the size of the launcher and the achieved non-diffractive range, the method has no particular limitations in terms of the achievable non-diffractive range of the generated Bessel beam/X-wave.
To validate this claim, we synthesized a launcher capable of radiating a Bessel beam with the TE 01 field distribution at the aperture with a 13000λ 0 of non-diffractive range (k ρ /k = 0.0068) at a single frequency point. Fig. 8 presents the exact dimensions of the profile and its E y field distribution in the x0z plane. In this particular case, the computational process was too demanding to apply the procedure on a range of frequencies, since the length of the launcher is around 1250λ 0 and its radius is around 90λ 0 . However, based on previous results, we can assume that, other than computational issues, there should be no particular difficulties in synthesizing an X-wave launcher with such and even larger non-diffractive ranges. It is worth noting that structures of this size cannot be analyzed using other full-wave methods with available computational resources. It is also clear that the fabrication of such a device can only be attempted for quite short wavelengths.
To provide a better understanding of the capabilities of the proposed launchers, we compare the capabilities of the launcher from Section III to other solutions used to generate X-waves: a dielectric lens filling a standard horn antenna [23], a dielectric metamaterial filling a standard horn antenna [24], a radial parallel-plate waveguide antenna [25], and RLSA launchers [26]. Solutions [23] and [24] show good behavior in terms of FBW and the dispersion of the axicon angle θ. More precisely [23] has a 42% of FBW and 2 • dispersion of θ, and [24] has a 50% of FBW and 7 • dispersion of θ. On the other hand, the advantage of [25] and [26] is that the launchers are low-profile structures. However, their drawback is the dispersive nature of radial launchers, particularly [25] has a 24 • dispersion of θ in 22% of FBW, and [26] has a 35 • dispersion of θ in 33% of FBW.
In comparison with the other solutions, horns generated by our approach show good performance in terms of FBW and dispersion of θ (the horn given in Section IV has 44% of FBW with 5 • of dispersion). Unlike other solutions, the generated horns are full-metal structures, overcoming the problems related to dielectric lenses at higher frequencies. Table 1 presents a comparison of the performance of the aforementioned launchers, as presented in the literature. The main drawback of our procedure is that the resulting structure (a cascade of the Marié mode converter and the horn) can be considered electrically large, especially if longer non-diffractive ranges are required. Although this might pose a size issue if the launchers are used at lower frequencies, the converter in [47] can help to render the TE 01 launcher more compact.

VI. GENERATING TM X-WAVES
Finally, we discuss the capability of the described procedure to synthesize TM X-wave launchers. Intuitively, one may assume that the problem for TM X-wave launchers is identical to the problem for TE X-wave launchers: synthesizing FIGURE 9. Distortion of a TM X-wave generated by a metallic horn antenna. (a) Normalized instantaneous intensity of H z in the x0z plane normalized to the wavelength at the central frequency of a TE X-wave with 44% of FBW within the non-diffractive range (b) Normalized instantaneous intensity of E z along the x0z plane normalized to the wavelength at the central frequency of a TM X-wave with 44% of FBW within the non-diffractive range in the same time slot t . The videos from which the pictures were extracted can be seen in the supplementary material of the paper. metal horns with aperture field distributions matching TM 0n modes of circular waveguides in a range of frequencies so that (7) is satisfied. However, this is not the case. The resulting radiated field distribution of such an aperture distribution indeed resembles X-wave distributions discussed in previous sections. On the other hand, it is heavily distorted compared to TE X-waves, as sketched in Fig. 9. This is most likely because TM 0n E field distributions in waveguides have much stronger edge components, resulting in high diffraction.
It is well-known that modal field distributions in metal waveguides and perfect magnetic conductor (PMC) waveguides are dual: field distributions of TE modes in metal waveguides match the TM field distributions in PMC, and vice versa. For instance, we can use the same horns used  for synthesizing TE X-waves, with PMC replacing the metal walls of the horn, and feed them with, for instance, a TM 01 mode using the mode-converter presented in Section IV-A of [49]. For demonstration purposes, we replaced the metal walls of the horn radiating a TE 02 X-wave from Section IV with corrugations to simulate PMC and fed the structure with a TM 01 mode. Corrugations were designed according to [29] and the realized corrugated profile and normalized dimensions of these corrugations are shown in Fig. 10.
The expected performance of the TM launcher designed with an ideal PMC is identical to the performance of the TE launcher. However, since we use corrugations instead of an ideal PMC, small aberrations are expected. Fig. 11 presents the instantaneous intensity of the E z in a time slot t max in which the intensity of the pulse is at its maximum value. Comparing Fig. 11 to Fig. 6(d) it is evident that the performance of the two launchers is nearly identical. In Fig. 12, by duality, we compare the E z field profile of the TM 02 launcher to the H z field profile of the TE 02 launcher along the x-axis 9.1λ 0 above the aperture at a single frequency point. The excellent agreement of the profiles verifies that TM X-wave launchers synthesized by replacing the metal walls of TE X-wave launchers with PMC indeed have identical performance as their TE counterparts.

VII. CONCLUSION
In this paper, a novel method for synthesizing X-wave launchers by circular metallic horns has been presented. X-waves are radiated by an axisymmetric TE (or TM ) field distribution on an electrically large aperture. The method has been implemented using an ad-hoc tool based on mode-matching framework. Launchers generated by the design procedure are full-metal horns, that present a low dispersion over a wide frequency range (more than 40% of FBW). We presented an example of a horn with 5 • of dispersion of the axicon angle θ in a 44% of FBW. We have also demonstrated that the procedure can synthetize Bessel launchers with arbitrary non-diffractive ranges. In particular, we proposed a Bessel beam launcher of 13000λ of non-diffractive range. Moreover, we have shown that the ratio between the non-diffractive range of the beam and the minimal length of the launcher capable of radiating such a beam is almost linear. The designed launchers do not require dielectric material or focusing elements to radiate X-waves and are also scalable at any frequency, making them particularly suited to the millimeter and submillimeter wave regimes. The proposed method and generated tool pave the way for many practical applications of X-waves in the submillimeter bands, such as imaging and near-field wireless communications. He has been involved in more than 60 research projects at the national and European levels and has co-supervised 23 Postdoctoral Fellows, 44 Ph.D. students, and 50 master's students. He has received 17 patents and is the author or coauthor of more than 250 journal articles and 510 publications in international conferences and workshops. He has shared the responsibility of the research activities on antennas at IETR in 2010 and 2011. His current research interests include numerical modeling (mainly FDTD), millimeter-wave printed and reconfigurable (MEMS) antennas, substrate integrated waveguide antennas, lens-based focusing devices, periodic and non-periodic structures (electromagnetic bandgap materials, metamaterials, reflectarrays, and transmitarrays), and biological effects of millimeter waves.