Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses
Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses
Khonina, Svetlana N.;Kharitonov, Sergey I.;Volotovskiy, Sergey G.;Soifer, Viktor A.
2021-07-05 00:00:00
hv photonics Article Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses 1 , 2 , 1 , 2 1 1 , 2 Svetlana N. Khonina * , Sergey I. Kharitonov , Sergey G. Volotovskiy and Viktor A. Soifer IPSI RAS—Branch of the FSRC “Crystallography and Photonics” RAS, Molodogvardeyskaya 151, 443001 Samara, Russia; prognoz@ipsiras.ru (S.I.K.); sv@ipsiras.ru (S.G.V.); soifer@ssau.ru (V.A.S.) Samara National Research University, Moskovskoye Shosse 34, 443086 Samara, Russia * Correspondence: khonina@ipsiras.ru; Tel./Fax: +7(846)332-56-20 Abstract: In this paper, we consider the comparative formation of perfect optical vortices in the non-paraxial mode using various optical elements: non-paraxial and parabolic toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements, as well as the calculation of caustic surfaces, is carried out using a hybrid geometrical-optical and wave approach. Numerical analysis performed on the basis of the expansion in conical waves qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions of the ring radius dependence on the order of the vortex phase singularity are analyzed. In the non-paraxial mode, when small light rings (several tens of wavelengths) are formed, a linear dependence of the ring radius on the vortex order is shown. The revealed features should be taken into account when using the considered optical elements forming the POV in various applications. Keywords: caustics; perfect optical vortices; toroidal vortex lens Citation: Khonina, S.N.; Kharitonov, S.I.; Volotovskiy, S.G.; Soifer, V.A. Caustics of Non-Paraxial Perfect Optical Vortices Generated by 1. Introduction Toroidal Vortex Lenses. Photonics Recently, the attention of researchers has been attracted by the “perfect” optical 2021, 8, 259. https://doi.org/ vortices (POVs) having a ring radius independent of its vortex number [1–4]. It is well 10.3390/photonics8070259 known that classical beams such as Laguerre–Gaussian beams [5–8] and higher-order Bessel beams [9–13] have a central light ring, the size of which is varied with the vortex Academic Editor: Federico Nguyen number. This feature may be undesirable in some applications, for example, when coupling different vortex beams into a fiber with a fixed annular profile [14]. Therefore, the main Received: 26 May 2021 advantage of POVs over other vortex beams is precisely in the fixed radius of the light Accepted: 29 June 2021 ring. Note that recently, various modifications of POVs have appeared, which do not Published: 5 July 2021 only have a ring structure. For example, elliptical POVs [14–18] in the form of different curves [19–23] and arrays [24–28], POVs with fractional optical vortex [29,30], as well as Publisher’s Note: MDPI stays neutral vector POVs [31–36]. with regard to jurisdictional claims in POVs are also used for optical capture and manipulation of microparticles [2,3,37,38] published maps and institutional affil- for free-space and underwater optical communication [39–41], for high-resolution plasmonic- iations. structured illumination microscopy [42], in the study of the noncollinear interaction of photons having orbital angular momentum (OAM) in spontaneous parametric down- conversion processes [43], for laser surface structuring [44], and for rotation speed detection of a spinning object based on the rotational Doppler effect [45]. Copyright: © 2021 by the authors. As a rule, the Fourier transformation of Bessel beams or lens-axicon doublets [4,43,46–49] Licensee MDPI, Basel, Switzerland. is used to generate such optical beams. In Reference [49], a comparison of POV generation This article is an open access article by means of different elements was investigated as follows: using a combination of a lens distributed under the terms and with an amplitude-phase element with a transmission function proportional to a Bessel conditions of the Creative Commons function, an optimal phase element with a transmission equal to the sign function of a Attribution (CC BY) license (https:// Bessel function, and a spiral axicon. In fact, these elements are similar, since the axicon is creativecommons.org/licenses/by/ 4.0/). Photonics 2021, 8, 259. https://doi.org/10.3390/photonics8070259 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 259 2 of 20 often used to generate Bessel beams [50–52]. A different approach for POV generation was suggested in [26,27] using curved fork gratings. In this paper, we consider another type of optical element, namely a toroidal lens that corresponds to a non-paraxial lens with radial displacement. The toroidal lens, instead of focusing to a point, focuses the incident radiation into a light ring [53,54]. Thus, the toroidal lens acts similarly to a lens-axicon doublet; however, it has certain advantages since it avoids the aberration problems associated with axicon’s tip fabrication [55]. Recently, the attention of researchers has been attracted by the toroidal wave front, which is studied in both the framework of the paraxial wave theory [56] and using the geometrical-optical approach [57,58]. The vortex toroidal lens, as well as the vortex axicon combined with a classic lens, allows for the formation of POVs. Note, if the ring formed in the focal plane has a small radius, then at large orders of the optical vortex, the POV ceases to be “perfect”. It was shown in [59] that for a POV there is a dependence on the order of the optical vortex, especially for optical systems with a low numerical aperture. A similar effect was noted in another work [60]. In this paper, we consider the formation of POV in a non-paraxial mode using toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements is carried out on the basis of a hybrid geometrical-optical and wave approach [20,60–62]. The asymptotic method for calculating the Kirchhoff integral is based on the geometric-optical approach with a finite (non-zero) ray thickness. This makes it possible to detect not only geometrical-optical caustics, but also areas of high intensity. Non-paraxial numerical analysis performed on the basis of the expansion in conical waves [63–65] qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions for the dependence of the ring radius on the order of the vortex phase singularity are analyzed. The obtained results can be useful in various applications using non-paraxial POVs, such as optical trapping and manipulation, vortex-based multiplexing, and laser structuring. 2. Parametric System of Equations for Calculating a 3D-Caustic Surface In optics, a caustic is the envelope of light rays reflected or refracted by a curved surface or object [66,67]. The main property of caustic surfaces (or lines) is that near these surfaces the intensity of the light field increases sharply (in the approximation of geometric optics, the intensity tends to infinity) [68,69]. Caustics connected with a curvature of the filed wavefront provide understanding to how the light redistribution evolves [70–73]. Therefore, caustics are used to analyze the features of structured laser beams, such as non-diffracting beams of various types [74–79], generalized Gaussian beams [80,81], accelerating and autofocusing beams [82–86], and vortex beams [20,61,62,87]. A general representation of the caustic surface was obtained for vortex optical ele- ments, the eikonal function of which can be represented in a separable form: F(r, q) = P(r) + q (1) where (r, q) are polar coordinates, k = 2p/l is the wavenumber of laser radiation with the wavelength l, and m is the order of the vortex phase singularity. Calculation of the Kirchhoff integral by the stationary phase method [20,61,62] leads to a parametric equation for calculating the 3D-caustic surface: 2 2 r(r) = A (r) + B (r), j(r, q) = q + tan [B(r)/A(r)], (2) 2 2 2 z(r, q) = S (r) + 2r r(r) cos[q j(r, q)] r r (r) . Photonics 2021, 8, 259 3 of 20 where A(r) = r + P (r)S(r), (3) S(r) B(r) = , where P (r) is the derivative of the radial term P(r) of the eikonal function (1), and the function S(r) is the solution to the quadratic equation: aS (r) + bS(r) + c = 0 (4) with coefficients determined by the following expressions: a = r P (r)P (r) , r rr h k i 3 2 2 b = r rP (r) + 1 P (r) P (r) + 2rP (r) r P (r) , rr r r rr r (5) h i 4 2 2 c = r 1 P (r) r . where P (r) is the second derivative of P(r). rr As follows from the above expressions, the effect of the vortex singularity is noticeable only if the ratio m / k is not too small, i.e., the value of the optical vortex m is commensurate with the wave number k. For conventional optical elements (several millimeters in size) used for the visible wavelength range, k is quite large (has a value of several thousand); therefore, the effect of the vortex singularity manifests itself only at very large values of m. It is this fact that determines the existence of the “perfect” optical vortices. However, if we consider microelements (several microns in size), then the effect of a vortex singularity with the value of m in several tens is already significant. In this work, for a clear demonstration of this effect, we consider microelements (i.e., elements with a size of several tens of microns). Further, we use the general formulas of this section to analyze different optical ele- ments, especially those that generate the POV. 3. Caustic Surface for Axisymmetric Optical Elements Forming a Light Ring Let us first consider axisymmetric optical elements that form an annular intensity distribution in a certain transverse plane. For axisymmetric optical elements (m = 0), caustic Equation (2) is simplified [88,89]: r(r) = r + P (r)S(r), (6) z(r) = S (r) [r r(r)] , where S(r) is one of the solutions of quadratic Equation (4): b b 4ac S (r) = , 2a a = r P (r)P (r), r rr h i (7) 3 2 b = r rP (r) + 1 P (r) P (r) , rr r h i 4 2 c = r 1 P (r) . After simplifications, instead of Equation (7), we obtain: h i h i 2 2 rP (r) + 1 P (r) P (r) rP (r) 1 P (r) P (r) rr r rr r r r S (r) = . (8) 2P (r)P (r) r rr One solution in Equation (8) corresponds to an off-axis caustic, and the second one corresponds to the axial caustic. Photonics 2021, 8, 259 4 of 20 In particular, the “+” sign corresponds to the off-axis caustic: 1 P (r) S (r) = (9) P (r) rr in which the surface is described by the following parametric equation: h i 2 1 r (r) = r + P (r)S (r) = r 1 P (r) P (r)P (r), + r + r r rr h i (10) 3/2 : 2 2 1 z (r) = S (r) [r r (r)] = 1 P (r) P (r). + + r rr The sign “