# A Leaky-Wave Analysis of Resonant Bessel-Beam Launchers: Design Criteria, Practical Examples, and Potential Applicationsat Microwave and Millimeter-Wave Frequencies

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

## 3. Leaky-Wave Analysis

#### 3.1. The Leaky-Wave Approach

#### 3.2. Design Parameters

#### 3.3. Dispersion Analysis

## 4. Structure Design

#### 4.1. PRS Implementation

#### 4.2. Feeder

#### 4.3. Comparison between Theoretical and Full-Wave Results

## 5. Applications

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

LWA | Leaky-Wave Antenna |

PCB | Printed-Circuit Board |

PEC | Perfect Electric Conductor |

PRS | Partially Reflecting Sheet |

TE | Transverse Electric |

TM | Transverse Magnetic |

VED | Vertical Electric Dipole |

VMD | Vertical Magnetic Dipole |

WPT | Wireless Power Transfer |

## References

- Hernández-Figueroa, H.E.; Zamboni-Rached, M.; Recami, E. Localized Waves; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Hernández-Figueroa, H.E.; Zamboni-Rached, M.; Recami, E. Nondiffracting Waves; John Wiley & Sons: Weinheim, Germany, 2013. [Google Scholar]
- McGloin, D.; Dholakia, K. Bessel beams: Diffraction in a new light. Contemporary Phys.
**2005**, 46, 15–28. [Google Scholar] [CrossRef] - Ettorre, M.; Pavone, S.C.; Casaletti, M.; Albani, M.; Mazzinghi, A.; Freni, A. Near-field focusing by non-diffracting Bessel beams. In Aperture Antennas for Millimeter and Sub-Millimeter Wave Applications; Springer: Cham, Switzerland, 2018; pp. 243–288. [Google Scholar]
- Ettorre, M.; Grbic, A. Generation of propagating Bessel beams using leaky-wave modes. IEEE Trans. Antennas Propag.
**2012**, 60, 3605–3613. [Google Scholar] [CrossRef] - Fuscaldo, W.; Valerio, G.; Galli, A.; Sauleau, R.; Grbic, A.; Ettorre, M. Higher-order leaky-mode Bessel-beam launcher. IEEE Trans. Antennas Propag.
**2016**, 64, 904–913. [Google Scholar] [CrossRef] [Green Version] - Comite, D.; Fuscaldo, W.; Podilchak, S.K.; Hílario-Re, P.D.; Gómez-Guillamón Buendía, V.; Burghignoli, P.; Baccarelli, P.; Galli, A. Radially periodic leaky-wave antenna for Bessel beam generation over a wide-frequency range. IEEE Trans. Antennas Propag.
**2018**, 66, 2828–2843. [Google Scholar] [CrossRef] - Chiotellis, N.; Mendez, V.; Rudolph, S.M.; Grbic, A. Experimental demonstration of highly localized pulses (X waves) at microwave frequencies. Phys. Rev. B
**2018**, 97, 085136. [Google Scholar] [CrossRef] [Green Version] - Chiotellis, N.; Zhang, S.; Vardaxoglou, Y.C.; Grbic, A. X wave radiator implemented with 3D printed metamaterials. IEEE Trans. Antennas Propag.
**2020**, 68, 5478–5486. [Google Scholar] [CrossRef] [Green Version] - Ettorre, M.; Rudolph, S.M.; Grbic, A. Generation of propagating Bessel beams using leaky-wave modes: Experimental validation. IEEE Trans. Antennas Propag.
**2012**, 60, 2645–2653. [Google Scholar] [CrossRef] - Fuscaldo, W.; Burghignoli, P.; Galli, A. Genealogy of leaky, surface, and plasmonic modes in partially open waveguides. Phys. Rev. Appl.
**2022**, 17, 034038. [Google Scholar] [CrossRef] - Pozar, D.M. Microwave Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2009. [Google Scholar]
- Lu, P.; Voyer, D.; Bréard, A.; Huillery, J.; Allard, B.; Lin-Shi, X.; Yang, X.S. Design of TE-polarized Bessel antenna in microwave range using leaky-wave modes. IEEE Trans. Antennas Propag.
**2017**, 66, 32–41. [Google Scholar] [CrossRef] - Lu, P.; Bréard, A.; Huillery, J.; Yang, X.S.; Voyer, D. Feeding coils design for TE-polarized Bessel antenna to generate rotationally symmetric magnetic field distribution. IEEE Antennas Wirel. Propag. Lett.
**2018**, 17, 2424–2428. [Google Scholar] [CrossRef] - Negri, E.; Fuscaldo, W.; Ettorre, M.; Burghignoli, P.; Galli, A. Analysis of resonant Bessel-beam launchers based on isotropic metasurfaces. In Proceedings of the 16th European Conference on Antennas and Propagation (EuCAP 2022), Madrid, Spain, 27 March–1 April 2022; pp. 1–4. [Google Scholar]
- Heebl, J.D.; Ettorre, M.; Grbic, A. Wireless links in the radiative near field via Bessel beams. Phys. Rev. Appl.
**2016**, 6, 034018. [Google Scholar] [CrossRef] [Green Version] - Negri, E.; Benassi, F.; Fuscaldo, W.; Masotti, D.; Burghignoli, P.; Costanzo, A.; Galli, A. Effective TE-polarized Bessel-beam excitation for wireless power transfer near-field links. In Proceedings of the 52nd European Microwave Conference (EuMC), Milan, Italy, 25–30 September 2022; pp. 1–4. [Google Scholar]
- Zamboni-Rached, M.; Recami, E.; Hernández-Figueroa, H.E. New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies. Eur. Phys. J. D
**2002**, 21, 217–228. [Google Scholar] [CrossRef] - Durnin, J.; Miceli, J.J., Jr.; Eberly, J.H. Diffraction-free beams. Phys. Rev. Lett.
**1987**, 58, 1499. [Google Scholar] [CrossRef] - Durnin, J. Exact solutions for nondiffracting beams. I. The scalar theory. J. Opt. Soc. Am. A
**1987**, 4, 651–654. [Google Scholar] [CrossRef] - Fuscaldo, W.; Pavone, S.C. Metrics for localized beams and pulses. IEEE Trans. Antennas Propag.
**2020**, 68, 1176–1180. [Google Scholar] [CrossRef] - Galli, A.; Baccarelli, P.; Burghignoli, P. Leaky-Wave Antennas. Wiley Enc. Electric. Electron. Eng.
**2016**, 2016, 1–20. [Google Scholar] - Tamir, T.; Oliner, A.A. Guided complex waves. Part 1: Fields at an interface. Proc. IEE
**1963**, 110, 310–324. [Google Scholar] [CrossRef] - Tamir, T.; Oliner, A.A. Guided complex waves. Part 2: Relation to radiation patterns. Proc. IEE
**1963**, 110, 325–334. [Google Scholar] [CrossRef] - Burghignoli, P.; Fuscaldo, W.; Comite, D.; Baccarelli, P.; Galli, A. Higher-order cylindrical leaky waves–Part I: Canonical sources and radiation formulas. IEEE Trans. Antennas Propag.
**2019**, 67, 6735–6747. [Google Scholar] [CrossRef] - Felsen, L. Real spectra, complex spectra, compact spectra. J. Opt. Soc. Am. A
**1986**, 3, 486–496. [Google Scholar] [CrossRef] - Luukkonen, O.; Simovski, C.; Granet, G.; Goussetis, G.; Lioubtchenko, D.; Raisanen, A.V.; Tretyakov, S.A. Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches. IEEE Trans. Antennas Propag.
**2008**, 56, 1624–1632. [Google Scholar] [CrossRef] - Fuscaldo, W.; Tofani, S.; Zografopoulos, D.C.; Baccarelli, P.; Burghignoli, P.; Beccherelli, R.; Galli, A. Systematic design of THz leaky-wave antennas based on homogenized metasurfaces. IEEE Trans. Antennas Propag.
**2018**, 66, 1169–1178. [Google Scholar] [CrossRef] - Benassi, F.; Fuscaldo, W.; Masotti, D.; Galli, A.; Costanzo, A. Wireless power transfer in the radiative near-field through resonant Bessel-beam launchers at millimeter Waves. In Proceedings of the 2021 IEEE Wireless Power Transfer Conference (WPTC), San Diego, CA, USA, 1–4 June 2021; pp. 1–4. [Google Scholar]
- Fuscaldo, W.; Galli, A.; Jackson, D.R. Optimization of 1-D unidirectional leaky-wave antennas based on partially reflecting sheets. IEEE Trans. Antennas Propag.
**2022**, 70, 7853–7868. [Google Scholar] [CrossRef] - Lovat, G.; Burghignoli, P.; Jackson, D.R. Fundamental properties and optimization of broadside radiation from uniform leaky-wave antennas. IEEE Trans. Antennas Propag.
**2006**, 54, 1442–1452. [Google Scholar] [CrossRef] - Sorrentino, R.; Mongiardo, M. Transverse resonance techniques. In Encyclopedia of RF and Microwave Engineering; John Wiley & Sons, Ltd: Hoboken, NJ, USA, 2005. [Google Scholar]
- Benassi, F.; Fuscaldo, W.; Negri, E.; Paolini, G.; Augello, E.; Masotti, D.; Burghignoli, P.; Galli, A.; Costanzo, A. Comparison between hybrid- and TM-polarized Bessel-beam launchers for wireless power transfer in the radiative near-field at millimeter waves. In Proceedings of the 51st European Microwave Conference (EuMC 2021), London, UK, 4–6 April 2022; pp. 1–4. [Google Scholar]
- Burghignoli, P.; Fuscaldo, W.; Galli, A. Fabry–Perot cavity antennas: The leaky-wave perspective. IEEE Antennas Propag. Mag.
**2021**, 63, 116–145. [Google Scholar] [CrossRef] - Xu, Y.; Peng, T.; Sun, M.; Luo, Y.; Wang, J.; Jiang, W.; Liu, G.; Wu, Z. Design and test of broadband rectangular waveguide TE
_{10}to circular waveguide TE_{21}and TE_{01}mode converters. IEEE Trans. Electron Devices**2019**, 66, 3573–3579. [Google Scholar] [CrossRef] - Chu, Q.X.; Mo, D.Y.; Wu, Q.S. An isolated radial power divider via circular waveguide TE
_{01}-mode transducer. IEEE Trans. Microw. Theory Tech.**2015**, 63, 3988–3996. [Google Scholar] [CrossRef] - Paković, S.; Zhou, S.; González-Ovejero, D.; Pavone, S.C.; Grbic, A.; Ettorre, M. Bessel–Gauss beam launchers for wireless power transfer. IEEE Open J. Antennas Propag.
**2021**, 2, 654–663. [Google Scholar] [CrossRef] - CST Studio Suite. Electromagnetic Field Simulation Software, Dassault Systèmes. Available online: https://www.3ds.com/products-services/simulia/products/cst-studio-suite/ (accessed on 1 October 2022).
- Balanis, C.A. Advanced Engineering Electromagnetics; Wiley: Hoboken, NJ, USA, 2012. [Google Scholar]
- Ruschin, S. Modified Bessel nondiffracting beams. J. Opt. Soc. Am. A
**1994**, 11, 3224–3228. [Google Scholar] [CrossRef] - Stsepuro, N.; Nosov, P.; Galkin, M.; Krasin, G.; Kovalev, M.; Kudryashov, S. Generating Bessel-Gaussian beams with controlled axial intensity distribution. Appl. Sci.
**2020**, 10, 7911. [Google Scholar] [CrossRef] - Abielmona, S.; Gupta, S.; Caloz, C. Compressive receiver using a CRLH-based dispersive delay line for analog signal processing. IEEE Trans. Microw. Theory Tech.
**2009**, 57, 2617–2626. [Google Scholar] [CrossRef] - Fuscaldo, W.; Benedetti, A.; Comite, D.; Baccarelli, P.; Burghignoli, P.; Galli, A. Bessel-Gauss beams through leaky waves: Focusing and diffractive properties. Phys. Rev. Appl.
**2020**, 13, 064040. [Google Scholar] [CrossRef] - Pavone, S.C.; Sorbello, G.; Di Donato, L. Improving physical optics approximation through Bessel beam scattering. IEEE Antennas Wirel. Propag. Lett.
**2021**, 20, 993–997. [Google Scholar] [CrossRef] - Chen, S.; Li, S.; Zhao, Y.; Liu, J.; Zhu, L.; Wang, A.; Du, J.; Shen, L.; Wang, J. Demonstration of 20-Gbit/s high-speed Bessel beam encoding/decoding link with adaptive turbulence compensation. Opt. Lett.
**2016**, 41, 4680–4683. [Google Scholar] [CrossRef] - Arlt, J.; Garcés-Chávez, V.; Sibbett, W.; Dholakia, K. Optical micromanipulation using a Bessel light beam. Opt. Commun.
**2001**, 197, 239–245. [Google Scholar] [CrossRef] - Lee, K.S.; Rolland, J.P. Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range. Opt. Lett.
**2008**, 33, 1696–1698. [Google Scholar] [CrossRef] [Green Version] - Planchon, T.A.; Gao, L.; Milkie, D.E.; Davidson, M.W.; Galbraith, J.A.; Galbraith, C.G.; Betzig, E. Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination. Nat. Methods
**2011**, 8, 417–423. [Google Scholar] [CrossRef] [Green Version] - Liu, C.; Zhao, Z.; Jin, C.; Xiao, Y.; Gao, G.; Xie, H.; Dai, Q.; Yin, H.; Kong, L. High-speed, multi-modal, label-free imaging of pathological slices with a Bessel beam. Biomed. Opt. Express
**2020**, 11, 2694–2704. [Google Scholar] [CrossRef] - Mugnai, D.; Spalla, P. Electromagnetic propagation of Bessel-like localized waves in the presence of absorbing media. Opt. Commun.
**2009**, 282, 4668–4671. [Google Scholar] [CrossRef] - Mazzinghi, A.; Balma, M.; Devona, D.; Guarnieri, G.; Mauriello, G.; Albani, M.; Freni, A. Large depth of field pseudo-Bessel beam generation with a RLSA antenna. IEEE Trans. Antennas Propag.
**2014**, 62, 3911–3919. [Google Scholar] [CrossRef] - Feng, R.; Ratni, B.; Yi, J.; Jiang, Z.; Zhang, H.; de Lustrac, A.; Burokur, S.N. Flexible manipulation of Bessel-like beams with a reconfigurable metasurface. Adv. Opt. Mat.
**2020**, 8, 2001084. [Google Scholar] [CrossRef] - Fuscaldo, W.; Pavone, S.C. Tunable terahertz Bessel-beam launchers. Rev. Electromagn.
**2022**, 1, 1–4. [Google Scholar] [CrossRef] - Bitman, A.; Moshe, I.; Zalevsky, Z. Improving depth-of field in broadband THz beams using nondiffractive Bessel beams. Opt. Lett.
**2012**, 37, 4164–4166. [Google Scholar] [CrossRef] [PubMed] - Winnerl, S.; Zimmermann, B.; Peter, F.; Schneider, H.; Helm, M. Terahertz Bessel-Gauss beams of radial and azimuthal polarization from microstructured photoconductive antennas. Opt. Express
**2009**, 17, 1571–1576. [Google Scholar] [CrossRef] [PubMed] - Monnai, Y.; Jahn, D.; Withayachumnankul, W.; Koch, M.; Shinoda, H. Terahertz plasmonic Bessel beamformer. Appl. Phys. Lett.
**2015**, 106, 021101. [Google Scholar] [CrossRef] - Yu, Y.Z.; Dou, W.B. Production of THz pseudo-Bessel beams with uniform axial intensity using irregular binary axicons. IET Optoelectron.
**2010**, 4, 195–200. [Google Scholar] [CrossRef] - Garnica, J.; Chinga, R.A.; Lin, J. Wireless power transmission: From far field to near field. Proc. IEEE
**2013**, 101, 1321–1331. [Google Scholar] [CrossRef] - Jolani, F.; Yu, Y.; Chen, Z. A planar magnetically coupled resonant wireless power transfer system using printed spiral coils. IEEE Antennas Wirel. Propag. Lett.
**2014**, 13, 1648–1651. [Google Scholar] [CrossRef] - Gowda, V.R.; Yurduseven, O.; Lipworth, G.; Zupan, T.; Reynolds, M.S.; Smith, D.R. Wireless power transfer in the radiative near field. IEEE Antennas Wirel. Propag. Lett.
**2016**, 15, 1865–1868. [Google Scholar] [CrossRef] - Fuscaldo, W.; Comite, D.; Boesso, A.; Beccarelli, P.; Burghignoli, P.; Galli, A. Focusing leaky waves: A class of electromagnetic localized waves with complex spectra. Phys. Rev. Appl.
**2018**, 9, 054005. [Google Scholar] [CrossRef] - Pavone, S.C.; Ettorre, M.; Casaletti, M.; Albani, M. Transverse circular-polarized Bessel beam generation by inward cylindrical aperture distribution. Opt. Express
**2016**, 24, 11103–11111. [Google Scholar] [CrossRef] - Pavone, S.C.; Ettorre, M.; Albani, M. Analysis and design of Bessel beam launchers: Longitudinal polarization. IEEE Trans. Antennas Propag.
**2016**, 64, 2311–2318. [Google Scholar] [CrossRef]

**Figure 1.**Pictorial representation of a TM- or TE-polarized resonant Bessel-beam launcher excited by a VED or a VMD with its $|{E}_{z}|$ or $|{H}_{z}|$ near-field distribution, respectively.

**Figure 2.**Design curves for a minimum ${z}_{\mathrm{ndr}}=15$ mm and a maximum ${\rho}_{\mathrm{ap}}=10$ mm. The normalized nondiffractive range ${\overline{z}}_{\mathrm{ndr}}={z}_{\mathrm{ndr}}/{\lambda}_{0}$ and the normalized aperture radius ${\overline{\rho}}_{\mathrm{ap}}={\rho}_{\mathrm{ap}}/{\lambda}_{0}$ are reported on the y-axis and the x-axis, respectively. Gray dashed lines represent the design boundaries at 30, 60, and 90 GHz, and differently colorized solid or dashed lines represent the design curves in a TM or TE polarization, respectively, for different radial resonances.

**Figure 3.**(

**a**) Transverse equivalent network of a resonant Bessel-beam launcher, and (

**b**) the geometry of the PRS fishnet-like unit cell. The areas in which PEC is not present are filled by the dielectric inside the cavity, constituted by air in this case. (

**c**) Dispersion diagrams of the normalized phase constant ${\widehat{\beta}}_{\rho}$ vs. frequency f for the specific case derived in Section 4, with the design parameters reported in Table 2. Red and blue solid lines represent the dispersion curves of TE and TM leaky modes, respectively. The radial resonances for TE and TM modes, represented by red and blue dashed lines, are obtained by enforcing a null of the first- or the zeroth-order Bessel function, respectively.

**Figure 4.**Full-wave and radiation-integral results for the longitudinal electric-field component ${E}_{z}$ in the TM-polarized Bessel-beam launcher on the left and on the right, respectively. At the top-left corner, the $|{E}_{z}|$ field distribution is represented over the transverse $xy$-plane at $z={z}_{\mathrm{ndr}}/2$.

**Figure 5.**Full-wave and radiation-integral results for the longitudinal magnetic-field component ${H}_{z}$ in the designed TE-polarized Bessel-beam launcher on the left and on the right, respectively. At the top-left corner, the $|{H}_{z}|$ field distribution is represented over the transverse $xy$-plane at $z={z}_{\mathrm{ndr}}/2$.

**Figure 6.**(

**a**) Pictorial representation of a radiative near-field WPT link between two resonant Bessel-beam launchers. (

**b**) The $|{S}_{21}|$ values in a radiative near-field WPT link at 90 GHz for different distances between transmitting and receiving TM-polarized Bessel-beam launchers.

Admittance | TE | TM |
---|---|---|

${Y}_{0}$ | $\frac{{k}_{z}}{{k}_{0}{\eta}_{0}}$ | $\frac{{k}_{0}}{{k}_{z}{\eta}_{0}}$ |

${Y}_{1}$ | $\frac{{k}_{z1}}{{k}_{0}{\eta}_{0}}$ | $\frac{{k}_{0}{\epsilon}_{\mathrm{r}}}{{k}_{z1}{\eta}_{0}}$ |

**Table 2.**Design parameters and radial wavenumbers for TE- and TM-polarized resonant Bessel-beam launchers with a working frequency ${f}_{0}=90$ GHz and an aperture radius ${\rho}_{\mathrm{ap}}=10$ mm. The period of the PRS unit cell is $p\simeq 3.33$ mm.

Polarization | ${\widehat{\mathit{\beta}}}_{\mathit{\rho}}$ | ${\widehat{\mathit{\alpha}}}_{\mathit{\rho}}$ | ${\mathit{X}}_{\mathit{s}}$$\left(\mathsf{\Omega}\right)$ | h (mm) | $\mathit{g}/\mathit{p}$ | $\mathit{w}/\mathit{p}$ |
---|---|---|---|---|---|---|

TE | 0.5401 | 0.0019 | 30 | 1.94 | 0.2086 | 0.05 |

TM | 0.4588 | 0.0020 | 20 | 1.84 | 0.1732 | 0.1 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Negri, E.; Fuscaldo, W.; Burghignoli, P.; Galli, A.
A Leaky-Wave Analysis of Resonant Bessel-Beam Launchers: Design Criteria, Practical Examples, and Potential Applicationsat Microwave and Millimeter-Wave Frequencies. *Micromachines* **2022**, *13*, 2230.
https://doi.org/10.3390/mi13122230

**AMA Style**

Negri E, Fuscaldo W, Burghignoli P, Galli A.
A Leaky-Wave Analysis of Resonant Bessel-Beam Launchers: Design Criteria, Practical Examples, and Potential Applicationsat Microwave and Millimeter-Wave Frequencies. *Micromachines*. 2022; 13(12):2230.
https://doi.org/10.3390/mi13122230

**Chicago/Turabian Style**

Negri, Edoardo, Walter Fuscaldo, Paolo Burghignoli, and Alessandro Galli.
2022. "A Leaky-Wave Analysis of Resonant Bessel-Beam Launchers: Design Criteria, Practical Examples, and Potential Applicationsat Microwave and Millimeter-Wave Frequencies" *Micromachines* 13, no. 12: 2230.
https://doi.org/10.3390/mi13122230