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Photonics
, Volume 8 (8) – Jul 28, 2021

/lp/multidisciplinary-digital-publishing-institute/toroidal-vortices-of-energy-in-tightly-focused-second-order-ZefjIWfz4i

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hv photonics Communication Toroidal Vortices of Energy in Tightly Focused Second-Order Cylindrical Vector Beams 1 , 2 , 1 , 2 1 , 2 Sergey S. Stafeev * , Elena S. Kozlova and Victor V. Kotlyar IPSI RAS—Branch of the FSRC “Crystallography and Photonics” RAS, 443001 Samara, Russia; kozlova.elena.s@gmail.com (E.S.K.); kotlyar@ipsiras.ru (V.V.K.) Department of Technical Cybernetics, Samara National Research University, 443086 Samara, Russia * Correspondence: sergey.stafeev@gmail.com Abstract: In this paper, we simulate the focusing of a cylindrical vector beam (CVB) of second order, using the Richards–Wolf formula. Many papers have been published on focusing CVB, but they did not report on forming of the toroidal vortices of energy (TVE) near the focus. TVE are ﬂuxes of light energy in longitudinal planes along closed paths around some critical points at which the ﬂux of energy is zero. In the 3D case, such longitudinal energy ﬂuxes form a toroidal surface, and the critical points around which the energy rotates form a circle lying in the transverse plane. TVE are formed in pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). We show that when light with a wavelength of 532 nm is focused by a lens with numerical aperture NA = 0.95, toroidal vortices periodically appear at a distance of about 0.45 m (0.85 ) from the axis (with a period along the z-axis of 0.8 m (1.5 )). The vortices arise in pairs: the vortex nearest to the focal plane is twisted clockwise, and the next vortex is twisted counterclockwise. These vortices are accompanied by saddle points. At higher distances from the z-axis, this pattern of toroidal vortices is repeated, and at a distance of about 0.7 m (1.3 ), a region in which toroidal vortices are repeated along the z-axis is observed. When the beam is focused and limited by a narrow annular aperture, these toroidal vortices are not observed. Citation: Stafeev, S.S.; Kozlova, E.S.; Kotlyar, V.V. Toroidal Vortices of Keywords: tight focusing; Richards–Wolf formula; cylindrical vector beam; energy backﬂow; toroidal Energy in Tightly Focused vortices; V-points Second-Order Cylindrical Vector Beams. Photonics 2021, 8, 301. https://doi.org/10.3390/ photonics8080301 1. Introduction Recently, the tight focusing of light with inhomogeneous polarization has attracted Received: 2 June 2021 attention from researchers, due to the wide variety of focal spot shapes that can be ob- Accepted: 25 July 2021 tained in this polarization state. For example, in previous work, compact foci have been Published: 28 July 2021 obtained with sizes smaller than the diffraction limit [1], as well as optical needles [2], light tunnels [3,4], chains of foci [5,6], and foci with a ﬂat apex [7,8]. Publisher’s Note: MDPI stays neutral Predominantly, the behavior of the intensity at the focus is investigated in the papers, with regard to jurisdictional claims in for instance, in [1–8], while the behavior of the energy ﬂux (Poynting vector) of focused published maps and institutional afﬁl- beams with inhomogeneous polarization is considered much less often [9–19]. For example, iations. the authors of [9] explored the tight focusing of a vector beam with a polarization that periodically changed from linear to circular along the radial or azimuthal axis. It was shown that there are points in the focus plane of these beams around which the Poynting vector rotates. The behavior of the Poynting vector in a tightly focused optical vortex with radial Copyright: © 2021 by the authors. and azimuthal polarization was investigated in [10] and [11], respectively. The inﬂuence of Licensee MDPI, Basel, Switzerland. the sector aperture on the energy ﬂux at the tight focus of an azimuthally polarized beam This article is an open access article was studied in [12], and in the same way as in [9], rotation of the Poynting vector around distributed under the terms and several points located along a certain circle was observed in the focal plane. The authors conditions of the Creative Commons of [13] investigated the energy ﬂux arising at the tight focus of a beam with polarization Attribution (CC BY) license (https:// close to azimuthal, but with an additional periodically changing radial component of creativecommons.org/licenses/by/ insigniﬁcant size (a kaleidoscope-structured vector optical beam). In [14], the self-healing 4.0/). Photonics 2021, 8, 301. https://doi.org/10.3390/photonics8080301 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 301 2 of 9 properties of Bessel–Gauss beams were studied, and in particular, the transverse component of the Poynting vector in beam sections was analyzed. It has previously been shown that in tightly focused beams with a polarization sin- gularity, there are regions in which the direction of the Poynting vector is opposite to the direction of the beam propagation [15,16]. The existence of such regions was noted quite a long time ago in the classical work of Richards and Wolf [20], and even earlier in the study of Ignatovsky [21]. However, the regions of negative values of the Poynting vector projection on the optical axis observed by them were small. The same areas are observed in the case of the light ﬁeld diffraction at the edges [22] and with total internal reﬂection [23,24]. However, it is possible to make such regions sufﬁciently large only while sharp focusing of beams with a phase or polarization singularity [15,16]. The study in [17] showed that in the Weber modes, the projection of the Poynting vector onto the optical axis takes negative values. The energy ﬂux is the sum of two ﬂuxes: the spin ﬂux and the orbital energy ﬂux [25,26]. Thus, the presence of areas in which the total energy ﬂux is directed towards the radiation source can be explained by the fact that the spin ﬂux directed towards the source exceeds in absolute value the orbital energy ﬂux [27]. A negative orbital energy ﬂux was observed in [26]. In an earlier paper [18], we found that regions of reverse energy ﬂux arise not only in the focus plane, but also in planes close to it. It should be also noted that the reverse energy ﬂow allows to control some types of microparticles in tasks of optical manipulation [25]. A microparticle placed in the backﬂow area will move towards the direction of light propagation. This phenomenon is called a tractor beam [28]. In this paper, using the Richards–Wolf formula we simulate the focusing of a cylin- drical vector beam (CVB) of second order. It is shown that toroidal vortices periodically appear at a distance of about 0.45 m from the axis (with a period along the z-axis of 0.8 m) when the light is focused by a lens with numerical aperture NA = 0.95. These vortices arise in pairs: the nearest to the focal plane vortex rotates clockwise, while the neighboring vortex rotates counterclockwise. The vortices are also accompanied by saddle points. The pattern of toroidal vortices is repeated at further distances from the z-axis. We observe a region in which toroidal vortices are repeated along the z-axis at a distance of about 0.7 m. These toroidal vortices are not observed when the beam is focused and limited by a narrow annular aperture. 2. Methods Our analysis relies on the Richards–Wolf integral [20], as a 2p max R R i f U(r, y, z) = B(q, j)T(q)P(q, j) (1) min 0 expfik[r sin q cos(j y) + z cos q]g sin q dq dj, where U(r, , z) is the electrical or magnetic ﬁeld in the focal spot; B(q, ') is the incident electrical or magnetic ﬁeld (where is the polar angle and ' is the azimuthal angle); T(q) 1/2 is the apodization function (the apodization function is equal to T(q) = cos q for an 3/2 aplanatic lens, whereas it is equal to T(q) = cos q for a ﬂat diffractive lens); k = 2/ is the wavenumber; is the wavelength; is the maximal polar angle determined by the max numerical aperture of the lens (NA = sin ); and P(q,') is the polarization matrix for max the electric and magnetic ﬁelds as 2 3 2 3 1 + cos j(cos q 1) sin j cos j(cos q 1) 4 5 4 5 P(q, j) = sin j cos j(cos q 1) a(q, j) + 1 + sin j(cos q 1) b(q, j), (2) sin q cos j sin q sin j where a(q,') and b(q,') are the polarization functions for the x- and y-components of the incident beam, respectively. Photonics 2021, 8, 301 3 of 9 th For a light ﬁeld with cylindrical polarization of the m order, the Jones vectors are a(q, f) sin(mf) E(q, f) = = , (3) b(q, f) cos(mf) for the electric ﬁeld, and a(q, f) cos(mf) H(q, f) = = , (4) b(q, f) sin(mf) for the magnetic ﬁeld. In [20], it was shown that combining Equation (1) with Equations (2)–(4) gives m+1 E = i [sin mjI + sin(m 2)jI ], x 0,m 2,m2 m+1 E = i [ cos mjI + cos(m 2)jI ], y 0,m 2,m2 E = 2i sin(m 1)jI , 1,m1 (5) m+1 H = i [cos mjI + cos(m 2)jI ], x 0,m 2,m2 m+1 H = i [sin mjI sin(m 2)jI ], y 0,m 2,m2 H = 2i cos(m 1)jI , 1,m1 where max ikz cos q I = A sin qT(q)(1 + cos q)A (q)e J (x) dq , 0,m m m min max ikz cos q I = A sin qT(q)(1 cos q)A (q)e J (x) dq , (6) 2,m2 m m2 min max 2 ikz cos q I = A sin qT(q)A (q)e J (x) dq . 1,m1 m m min The longitudinal projection of the Poynting vector S = (S , S , S ,), S = Re(E H * x y z z x y E H *) can be written as y x S = [sin mjI + sin(m 2)jI ] [sin mjI sin(m 2)jI ] z 0,m 2,m2 0,m 2,m2 (7) [ cos mjI + cos(m 2)jI ] [cos mjI + cos(m 2)jI ] . 0,m 2,m2 0,m 2,m2 For an angle j = 0, this is equal to 2 2 S = I I , (8) 0,m 2,m2 We now consider the case where = /2, A() = ( /2). (9) This corresponds to a high numerical aperture lens (NA = 1) limited by a narrow annular aperture. We have previously shown [29] that the area of energy backﬂow can be enhanced in this case due to enlarging of the energy backﬂow depth, and the width is slightly decreased. Taking into account Equation (9), Equation (6) for I and I can be rewritten as 0,m 2,m2 I = AT q ! J (kr) , 0,m (10) I = AT q ! J (kr) . 2,m2 m2 The longitudinal component S for m = 2 is equal to h i 2 2 S = AT q ! J (kr) J (kr) (11) 2 0 The above equation shows that when the condition in Equation (9) is valid, the areas in which S takes on negative values do not depend on the z-coordinate. Note that the z Photonics 2021, 8, x FOR PEER REVIEW 4 of 10 IA=θT → J ()kρ , 0,mm (10) I =θ AT → J () kρ . 2,mm −− 2 2 The longitudinal component Sz for m = 2 is equal to SA=θT → J kρ −J kρ () ( ) (11) z 20 Photonics 2021, 8, 301 4 of 9 The above equation shows that when the condition in Equation (9) is valid, the areas in which Sz takes on negative values do not depend on the z-coordinate. Note that the apodization function is always positive and does not affect the sign change in Equation (11). apodization function is always positive and does not affect the sign change in Equation 2 2 Figure 1 shows a plot of the function J (kr) J (kr) for = 0.532 m. 2 2 0 2 (11). Figure 1 shows a plot of the function J2 (kρ)−J0 (kρ) for λ = 0.532 μm. 2 2 Figure 1. Plot of the function J2 2 (kρ) − J0 2 (kρ). Figure 1. Plot of the function J (kr) J (kr). 2 0 Figure 1 shows that the longitudinal projection of the Poynting vector can take neg- Figure 1 shows that the longitudinal projection of the Poynting vector can take negative ative values. Areas of negative values also arise periodically at further distances from the values. Areas of negative values also arise periodically at further distances from the axis. Figur axis. F e 1igu shows re 1 shows a m a maximum aximum poss possible ratio ible ra of 5:1 tio o between f 5:1 between the reverse ene the reverse energy ﬂow and rgy flow the and the forward energy flow in free space. forward energy ﬂow in free space. The The distrib distribution ution shown in shown in Figur Figue re 1 1 for forthe the case de case described scribed by E by Equation quation (9) (9) c can an be be consider consider ed ed as as a a continuation continuation of of the the study study in in [ 19 [1], 9]which , which shown tha shown that t r egions regions wi with th negative negative values of the Poynting vector arise periodically for interference between four plane waves values of the Poynting vector arise periodically for interference between four plane converging at a large angle. In the present case, there is interference between an inﬁnite waves converging at a large angle. In the present case, there is interference between an number infinite n ofu spherical mber of spher wavesical with wav continuous es with cont changes inuou in s ch polarization anges in polar instead izatof ion four instplane ead of waves. The areas containing negative values acquire an axisymmetric character and they four plane waves. The areas containing negative values acquire an axisymmetric char- are also periodically repeated. acter and they are also periodically repeated. Figure 2 shows the results of a simulation based on Equation (1), in which CVB acquire Figure 2 shows the results of a simulation based on Equation (1), in which CVB ac- Photonics 2021, 8, x FOR PEER REVIEW of second order is focused with a lens with numerical aperture NA = 0.99, limited5 of by 10 a quire of second order is focused with a lens with numerical aperture NA = 0.99, limited narrow annular aperture with = 0.9 = 0.9 asin(NA), m = 2. min max by a narrow annular aperture with αmin = 0.9 × αmax =0.9 × asin(NA), m = 2. Figure 2. Intensity and direction of the Poynting vectors in the YZ-plane of the focused CVB of Figure 2. Intensity and direction of the Poynting vectors in the YZ-plane of the focused CVB of second order using a lens with numerical aperture NA = 0.99, limited by a narrow annular aperture second order using a lens with numerical aperture NA = 0.99, limited by a narrow annular aperture with αmin = 0.9 × αmax. with = 0.9 . min max Figure 2 shows that the areas where the direction of the Poynting vector is reversed approximately coincide with similar areas in Figure 1. There are no toroidal energy vor- tices in this case. 3. Results Based on the Richards–Wolf formula in (1), we then simulated the focusing of a CVB of second order (m = 2) with a wavelength of 532 nm, using a wide-aperture lens without an annular aperture. In this simulation, we used a lens with numerical aperture NA = 0.95. Figure 3 shows the results of focusing a cylindrical second-order vector beam using a lens with numerical aperture NA = 0.95 in the region 0.5 μm < z < 3 μm, 0.3 μm < r < 1 μm. (a) (b) Photonics 2021, 8, x FOR PEER REVIEW 5 of 10 Figure 2. Intensity and direction of the Poynting vectors in the YZ-plane of the focused CVB of second order using a lens with numerical aperture NA = 0.99, limited by a narrow annular aperture with αmin = 0.9 × αmax. Photonics 2021, 8, 301 5 of 9 Figure 2 shows that the areas where the direction of the Poynting vector is reversed approximately coincide with similar areas in Figure 1. There are no toroidal energy vor- tices in this case. Figure 2 shows that the areas where the direction of the Poynting vector is reversed approximately coincide with similar areas in Figure 1. There are no toroidal energy vortices 3. Results in this case. Based on the Richards–Wolf formula in (1), we then simulated the focusing of a CVB 3. Results of second order (m = 2) with a wavelength of 532 nm, using a wide-aperture lens without an annular aperture. In this simulation, we used a lens with numerical aperture NA = Based on the Richards–Wolf formula in (1), we then simulated the focusing of a CVB 0.95. of second order (m = 2) with a wavelength of 532 nm, using a wide-aperture lens without Figure 3 shows the results of focusing a cylindrical second-order vector beam using an annular aperture. In this simulation, we used a lens with numerical aperture NA = 0.95. a lens with numerical aperture NA = 0.95 in the region 0.5 μm < z < 3 μm, 0.3 μm < r < 1 Figure 3 shows the results of focusing a cylindrical second-order vector beam using a μ lens m. with numerical aperture NA = 0.95 in the region 0.5 m < z < 3 m, 0.3 m < r < 1 m. Photonics 2021, 8, x FOR PEER REVIEW 6 of 10 (a) (b) (c) Figure 3. (a) Intensity distribution (color) and directions of the Poynting vector (arrows) in the YZ plane and (b) positions Figure 3. (a) Intensity distribution (color) and directions of the Poynting vector (arrows) in the YZ plane and (b) positions of the points at which the Poynting vector is zero (squares-vortex with clockwise rotation, triangles-vortex with counter- of the points at which the Poynting vector is zero (squares-vortex with clockwise rotation, triangles-vortex with counter- clockwise rotation arrows, circles-saddle points). The numbers indicate the individual areas shown in the figures below. clockwise rotation arrows, circles-saddle points). The numbers indicate the individual areas shown in the ﬁgures below. (с) Schematic representation of a toroidal vortex of energy. (c) Schematic representation of a toroidal vortex of energy. It can be seen from Figure 3 that there are regions in the longitudinal plane YZ around which the Poynting vector rotates. This area will have the form of a circle (‘dark ring’) in the transverse XY plane. The Poynting vector is equal to zero in it. The Poynting vector will rotate around this circle in the YZ plane (or any other longitudinal plane containing the optical axis) and the surface formed by the Poynting vector trajectories in different longitudinal planes will have the form of a torus (longitudinal cross-section of which can be either a circle or an ellipse). Figure 3c schematically shows such a toroidal vortex. From Figure 3b, it can be seen that toroidal vortices appear periodically at a distance of approximately 0.45 μm from the axis (with a period along the z-axis of 0.8 μm). These vortices arise in pairs: the nearest to the focal plane vortex rotates clockwise (Figure 4), while the neighboring vortex rotates counterclockwise (Figure 5) and is located at a dis- tance of approximately 0.3 μm from the clockwise vortex. The clockwise vortices are lo- cated slightly farther (about 0.05 μm) from the z-axis (r = 0) than the counterclockwise ones. The vortices are accompanied by saddle points (Figure 6), the closest of which are located at a distance of 0.4 μm from the axis. Note that the distance at which the toroidal vortices appear (the center of the vortices) approximately corresponds to the third zero of 2 2 the function J2 (kρ) − J0 (kρ) in Figure 1. Such saddle points as well as toroidal vortices form a dark ring in the XY plane. However, in the longitudinal section YZ the Poynting vector behaves differently near such points: the vector trajectories have the form of hy- perbolas (Figure 6). At further distances from the z-axis, the pattern of toroidal vortices is repeated: a region of repeating toroidal vortices along the z-axis is also observed at a distance of ap- proximately 0.7 μm. This distance approximately corresponds to the fifth zero of the 2 2 function J2 (kρ) − J0 (kρ) in Figure 1. Photonics 2021, 8, 301 6 of 9 It can be seen from Figure 3 that there are regions in the longitudinal plane YZ around which the Poynting vector rotates. This area will have the form of a circle (‘dark ring’) in the transverse XY plane. The Poynting vector is equal to zero in it. The Poynting vector will rotate around this circle in the YZ plane (or any other longitudinal plane containing the optical axis) and the surface formed by the Poynting vector trajectories in different longitudinal planes will have the form of a torus (longitudinal cross-section of which can be either a circle or an ellipse). Figure 3c schematically shows such a toroidal vortex. From Figure 3b, it can be seen that toroidal vortices appear periodically at a distance of approximately 0.45 m from the axis (with a period along the z-axis of 0.8 m). These vortices arise in pairs: the nearest to the focal plane vortex rotates clockwise (Figure 4), while the neighboring vortex rotates counterclockwise (Figure 5) and is located at a distance of approximately 0.3 m from the clockwise vortex. The clockwise vortices are located slightly farther (about 0.05 m) from the z-axis (r = 0) than the counterclockwise ones. The vortices are accompanied by saddle points (Figure 6), the closest of which are located at a distance of 0.4 m from the axis. Note that the distance at which the toroidal vortices appear (the center of the vortices) approximately corresponds to the third zero of the 2 2 function J (kr) J (kr) in Figure 1. Such saddle points as well as toroidal vortices form a 2 0 dark ring in the XY plane. However, in the longitudinal section YZ the Poynting vector behaves differently near such points: the vector trajectories have the form of hyperbolas Photonics 2021, 8, x FOR PEER REVIEW 7 of 10 (Figure 6). Photonics 2021, 8, x FOR PEER REVIEW 7 of 10 Figure 4. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region Figure 4. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region of the toroidal vortex that rotates clockwise and is nearest to the r- and z-axes (area 1 in Figure 3a). Figure 4. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region of the toroidal vortex that rotates clockwise and is nearest to the r- and z-axes (area 1 in Figure 3a). of the toroidal vortex that rotates clockwise and is nearest to the r- and z-axes (area 1 in Figure 3a). Figure 5. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region of the toroidal vortex that rotates counterclockwise and is nearest to the r- and z-axes (area 2 in Figure 5. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region Figure 5. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region Figure 3a). of the toroidal vortex that rotates counterclockwise and is nearest to the r- and z-axes (area 2 in of the toroidal vortex that rotates counterclockwise and is nearest to the r- and z-axes (area 2 in Figure 3a). Figure 3a). Figure 6. Intensity distribution (color) and direction of the Poynting vector (arrows) in the region of the saddle point nearest to the r- and z-axes (area 3 in Figure 3a). Figure 6. Intensity distribution (color) and direction of the Poynting vector (arrows) in the region of the saddle point nearest to the r- and z-axes (area 3 in Figure 3a). Figures 4–6 show the points of polarization singularity (V-points) at which the di- rection of linear polarization is undefined. The Poincaré–Hopf index is equal to η = +1 Figures 4–6 show the points of polarization singularity (V-points) at which the di- rection of linear polarization is undefined. The Poincaré–Hopf index is equal to η = +1 Photonics 2021, 8, x FOR PEER REVIEW 7 of 10 Figure 4. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region of the toroidal vortex that rotates clockwise and is nearest to the r- and z-axes (area 1 in Figure 3a). Photonics 2021, 8, 301 7 of 9 Figure 5. Intensity distribution (color) and directions of the Poynting vectors (arrows) in the region of the toroidal vortex that rotates counterclockwise and is nearest to the r- and z-axes (area 2 in Figure 3a). Figure 6. Intensity distribution (color) and direction of the Poynting vector (arrows) in the region of Figure 6. Intensity distribution (color) and direction of the Poynting vector (arrows) in the region of the saddle point nearest to the r- and z-axes (area 3 in Figure 3a). the saddle point nearest to the r- and z-axes (area 3 in Figure 3a). Figures 4–6 show the points of polarization singularity (V-points) at which the di- At further distances from the z-axis, the pattern of toroidal vortices is repeated: a rection of linear polarization is undefined. The Poincaré–Hopf index is equal to η = +1 region of repeating toroidal vortices along the z-axis is also observed at a distance of approximately 0.7 m. This distance approximately corresponds to the ﬁfth zero of the 2 2 function J (kr) J (kr) in Figure 1. 2 0 Figures 4–6 show the points of polarization singularity (V-points) at which the di- rection of linear polarization is undeﬁned. The Poincaré–Hopf index is equal to = +1 (Figures 4 and 5) and = –1 (Figure 6) [30]. In this research, we investigate V-points in the longitudinal plane, in contrast to the previous study [30], in which these points were investigated in the transverse plane. 4. Discussion A lot of papers sharply focused on CVB are known today [1,8,31,32]. However, as a rule, these studies considered the energy distribution of the energy ﬂux (projections of the Poynting vector) in the transverse regions near the focus and almost never considered energy ﬂows in the longitudinal (meridional) planes. In this research, using focusing of the second-order CVB as example, we have shown that many vortices of energy or toroidal vortices of energy (TVE) can be formed near the focus in the meridional planes. TVE are ﬂows of light energy in meridional planes along closed paths around some critical points at which the ﬂow of energy is zero. In the 3D case, such longitudinal energy ﬂuxes form the surface of the torus and the critical points around which the energy rotates form a circle lying in the transverse plane. TVEs are formed by pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). Figure 7 shows some vortices of different nature (polarization (a), phase (b), and energy (c) vortices). The topology of such vortex ﬂows is largely the same. The singularity index of V-points (Figure 7a) and the topological charge of the phase singularity (Figure 7b) are determined by summation of the number of the linear polarization vector complete revolutions [30] or the phase jumps by 2 [33] along a closed loop around the critical (singular) point. Therefore, the energy vortex in Figure 7c can be assigned the topological index 1 since the direction of the Poynting vector changed by 2 clockwise when going around the point of the energy ﬂux uncertainty. The vortex energy ﬂow index in Figure 5 is +1 since the critical point is bypassed counterclockwise. Photonics 2021, 8, x FOR PEER REVIEW 8 of 10 (Figures 4 and 5) and η = –1 (Figure 6) [30]. In this research, we investigate V-points in the longitudinal plane, in contrast to the previous study [30], in which these points were in- vestigated in the transverse plane. 4. Discussion A lot of papers sharply focused on CVB are known today [1,8,31,32]. However, as a rule, these studies considered the energy distribution of the energy flux (projections of the Poynting vector) in the transverse regions near the focus and almost never considered energy flows in the longitudinal (meridional) planes. In this research, using focusing of the second-order CVB as example, we have shown that many vortices of energy or to- roidal vortices of energy (TVE) can be formed near the focus in the meridional planes. TVE are flows of light energy in meridional planes along closed paths around some crit- ical points at which the flow of energy is zero. In the 3D case, such longitudinal energy fluxes form the surface of the torus and the critical points around which the energy ro- tates form a circle lying in the transverse plane. TVEs are formed by pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). Figure 7 shows some vortices of different nature (polarization (a), phase (b), and energy (c) vortices). The topology of such vortex flows is largely the same. The singular- ity index of V-points (Figure 7a) and the topological charge of the phase singularity (Figure 7b) are determined by summation of the number of the linear polarization vector complete revolutions [30] or the phase jumps by 2π [33] along a closed loop around the critical (singular) point. Therefore, the energy vortex in Figure 7c can be assigned the Photonics 2021, 8, 301 8 of 9 topological index −1 since the direction of the Poynting vector changed by 2π clockwise when going around the point of the energy flux uncertainty. The vortex energy flow in- dex in Figure 5 is +1 since the critical point is bypassed counterclockwise. (a) (b) (c) Figure 7. Examples of critical points (singularity points): (a) V-point at the center of the CVB field with order n = 3 (arrows Figure 7. Examples of critical points (singularity points): (a) V-point at the center of the CVB ﬁeld with order n = 3 (arrows show the directions of local linear polarization vectors); (b) phase (from zero to 2π) of an optical vortex with the center of show the directions of local linear polarization vectors); (b) phase (from zero to 2) of an optical vortex with the center of a a fifth-order phase singularity (topological charge is 5); and (c) the point of uncertainty of the energy flow direction in the ﬁfth-order phase singularity (topological charge is 5); and (c) the point of uncertainty of the energy ﬂow direction in the meridional plane at the focus of the CVB with the order n = 2 (Figure 4). meridional plane at the focus of the CVB with the order n = 2 (Figure 4). 5. Conclusions 5. Conclusions In this paper, the focusing of second-order CVBs was numerically simulated using In this paper, the focusing of second-order CVBs was numerically simulated using the Richards–Wolf formulas. The goal of the study was to determine the conditions for the Richards–Wolf formulas. The goal of the study was to determine the conditions for the formation of toroidal vortices of energy around the ‘dark rings’ (rings on which the the formation of toroidal vortices of energy around the ‘dark rings’ (rings on which the Poynting vector is zero), not only in the focal plane but also in planes near the focus. Poynting vector is zero), not only in the focal plane but also in planes near the focus. It was shown that when light is focused by a lens with numerical aperture NA = 0.95, It was shown that when light is focused by a lens with numerical aperture NA = 0.95, toroidal vortices periodically appear at a distance of about 0.45 μm from the axis (with a toroidal vortices periodically appear at a distance of about 0.45 m from the axis (with a period along the z-axis of 0.8 μm). These vortices arise in pairs: the one nearest to the period along the z-axis of 0.8 m). These vortices arise in pairs: the one nearest to the focal focal plane rotates clockwise, and the neighboring one rotates counterclockwise. They are plane rotates clockwise, and the neighboring one rotates counterclockwise. They are also also accompanied by saddle points. At further distances from the z-axis, the pattern of accompanied by saddle points. At further distances from the z-axis, the pattern of toroidal toroidal vortices repeats: at a distance of about 0.7 μm, a region in which toroidal vortices vortices repeats: at a distance of about 0.7 m, a region in which toroidal vortices repeats along the z-axis is observed. When the beam is focused and limited by a narrow annular aperture, toroidal vortices are not observed. It should be noted that a similar effect can be observed not only for polarization vortices, but also for phase optical vortices. In particular, this holds for optical vortices with a topological charge of two and circular polarization; in addition to the beams considered above, negative values of the longitudinal projection of the Poynting vector are observed in this case on the optical axis under tight focusing conditions. Author Contributions: Conceptualization, S.S.S. and V.V.K.; Methodology, S.S.S. and V.V.K.; Soft- ware, S.S.S. and E.S.K.; Validation, V.V.K.; Formal analysis, V.V.K.; Investigation, S.S.S., E.S.K., and V.V.K.; Resources, S.S.S.; Data curation, S.S.S.; Writing—original draft preparation, S.S.S.; Writing— review and editing, V.V.K.; Visualization, S.S.S. and E.S.K.; Supervision, V.V.K.; Project administration, V.V.K.; Funding acquisition, S.S.S. and V.V.K. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Russian Science Foundation, grant number 18-19-00595 (in part of “Materials and Methods”), by the Ministry of Science and Higher Education within the State as- signment FSRC “Crystallography and Photonics” RAS (in part of “Introduction” and “Conclusions”). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Code underlying the results presented in this paper is available in https://github.com/Sergey-St/richwolf (accessed on 20 May 2021). Conﬂicts of Interest: The authors declare no conﬂict of interest. Photonics 2021, 8, 301 9 of 9 References 1. Dorn, R.; Quabis, S.; Leuchs, G. Sharper focus for a radially polarized light beam. Phys. Rev. Lett. 2003, 91, 233901. [CrossRef] 2. Chong, C.T.; Sheppard, C.; Wang, H.; Shi, L.; Lukyanchuk, B. Creation of a needle of longitudinally polarized light in vacuum using binary optics. Nat. Photonics 2008, 2, 501–505. 3. 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Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Jul 28, 2021

**Keywords: **tight focusing; Richards–Wolf formula; cylindrical vector beam; energy backflow; toroidal vortices; V-points

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