Citation: Godana, Z.T.; Hebling, J.; Pálfalvi, L. Focusing of Radially Polarized Electromagnetic Waves by a Parabolic Mirror. Photonics 2023, 10, 848. https://doi.org/10.3390/ photonics10070848 Received: 29 June 2023 Revised: 17 July 2023 Accepted: 18 July 2023 Published: 21 July 2023 Copyright: © 2023 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/). photonics hv Communication Focusing of Radially Polarized Electromagnetic Waves by a Parabolic Mirror Zerihun Tadele Godana 1,2, *, János Hebling 1,2 and László Pálfalvi 1,3 1 Institute of Physics, University of Pécs, H-7624 Pécs, Hungary; hebling@gamma.ttk.pte.hu (J.H.); palfalvi@gamma.ttk.pte.hu (L.P.) 2 Szentágothai Research Centre, University of Pécs, H-7624 Pécs, Hungary 3 ELKH—PTE High-Field Terahertz Research Group, H-7624 Pécs, Hungary * Correspondence: godanaze@gamma.ttk.pte.hu or zerihunta@du.edu.et Abstract: It is well-known that a strong longitudinal electric field and a small spot size are observed when radially polarized beams are tightly focused using a high numerical aperture parabolic mirror. The longitudinal electric field component can accelerate electrons along the propagation axis at high intensities in the focal region, which opens an application in particle acceleration. In this paper, we present a rigorous derivation of the electric field obtained when a radially polarized, monochromatic, flat-top beam is focused by a parabolic mirror. The formulae were deduced from the Stratton–Chu integral known from vector diffraction theory. We examined the influence of the focusing parameters on the distribution of both the longitudinal and radial electric field components. In the small numerical aperture and short wavelength regimes, excellent agreement was found with the results obtained from the Rayleigh–Sommerfeld formula. The calculation method can be adapted for various beam types and for electromagnetic pulses as well. Keywords: focusing; parabolic mirror; radially polarized beams; Stratton–Chu integral 1. Introduction The paraboloid mirror can focus light nearly within a 4π solid angle [1,2]. Because of this special characteristic, researchers have been strongly interested in parabolic mirrors over the past several decades. The vector field nature of the light becomes essential in accurate description of a nonparaxial beam, when a beam is tightly focused by a parabolic mirror. The high-intensity laser community, which makes major efforts to achieve the highest laser intensities, is strongly interested in examinations of the vector field focusing characteristics [3]. By tightly focusing the beam in a vacuum using an off-axis parabolic (OAP) mirror, it is possible to attain extremely high intensity, which makes this of in- terest for laser-based particle acceleration [4]. For such a high-intensity, tightly focused electromagnetic field, a detailed description of the focused field is necessary in order to precisely identify the motion of charged particles [5]. Using a high numerical aperture parabolic mirror with a radially polarized beam is ideal for achieving a small focal spot size and strong longitudinal electric field, which opens an application possibility in particle acceleration [6,7]. Beginning with a study by Ignatovsky in 1920, a detailed diffraction theory of focused light from parabolic mirrors has been developed over the period of nearly a century. Ignatowsky transformed Maxwell’s equations to the parabolic coordinate system, set the boundary values, and then used these boundary values to solve Maxwell’s equations [8]. Richards and Wolf provided a different theoretical approach in which strongly focused beams were precisely defined in terms of the field distribution of the collimated input beam at the entry pupil of the focusing apparatus [9]. There are different approaches for evaluating tightly focused beams based on the Stratton–Chu formulation of Green’s Photonics 2023, 10, 848. https://doi.org/10.3390/photonics10070848 https://www.mdpi.com/journal/photonics