# Generation of Light Fields with Controlled Non-Uniform Elliptical Polarization When Focusing on Structured Laser Beams

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

_{x}, A

_{y}are amplitudes and δ

_{x}, δ

_{y}are phases of corresponding components, and Δ = δ

_{y}− δ

_{x}.

- the ratio of the semi-axes (which depends on amplitudes A
_{x}and A_{y}); - inclination of the semi-major axis (i.e., angle α);
- vector rotation direction.

## 3. Results

#### 3.1. Variable Tilt Angle of the Polarization Ellipse

_{x}and A

_{y}and consider an arbitrary function for a variable slope angle α(θ, φ). Then, the components of the electric field are expressed by the following formulas:

#### 3.1.1. The Tilt Angle of the Polarization Ellipse Is Equal to the Polar Angle

_{p}

_{,q}for p + q ≤ 3 (these will be needed here and below):

_{x}and A

_{y}, but ${S}_{z}$ in Equation (19) does not have such a dependence and takes everywhere real positive values.

_{x}= A

_{y}provides a completely uniform distribution of the polarization of the input field; however, in the focal plane, the polarization state becomes inhomogeneous (Figure 3a,b). The inequality A

_{x}≠ A

_{y}leads to inhomogeneity of the polarization distribution both in the input and in the focused field. In the example A

_{x}> A

_{y}(Figure 3c,d), according to Equation (16), there is an increase in the proportion of the longitudinal component.

#### 3.1.2. The Tilt Angle of the Polarization Ellipse Is a Multiple of the Polar Angle

_{x}= A

_{y}. The inequality A

_{x}≠ A

_{y}leads to a situation when the intensity will have (2p − 2) maxima and (2p − 2) minima (except for circles where ${J}_{p-1}^{2}(a)+{J}_{p}(a){J}_{p-2}(a)=0$).

_{x}= 2A

_{y}to A

_{x}= 0.5A

_{y}. When p = 2, there is practically a rotation of the entire distribution in the focal plane by 90 degrees (compare the last two lines in Figure 4). The case p = 2 is a special one since in this case some terms in Equation (23) are set to zero, so the structure is pretty simple. At p = 3, only the intensity distribution rotates by 45 degrees, and the polarization state changes in a more complex way (compare the last two lines in Figure 5).

_{x}= 2A

_{y}to A

_{x}= 0.5A

_{y}, a qualitative change occurs not only in the distribution of the polarization state but also in the distribution of the total intensity.

_{x}and A

_{y}, which was analytically shown for p = 1 in Equation (19).

#### 3.2. Variable Ratio of the Semi-Axes of the Polarization Ellipse

_{x}and A

_{y}ratio.

#### 3.2.1. Simple Trigonometric Dependence on the Polar Angle

_{p},

_{q}expressions are given in Equation (14).

#### 3.2.2. Multiple and Power Trigonometric Dependence on the Polar Angle

^{p}(φ) for odd p is expressed through sin(q

_{n}φ) where all q

_{n}are odd, and for even p, it is expressed through cos(q

_{n}φ) where all q

_{n}are even. Similarly, cos

^{p}(φ) is expressed in terms of cos(q

_{n}φ) where all q

_{n}have the same parity as p. Therefore, the power dependence provides a locally linear polarization in similar situations as for the multiple ones.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Forbes, A. Structured Light from Lasers. Laser Photon. Rev.
**2019**, 13, 1900140. [Google Scholar] [CrossRef] - Angelsky, O.V.; Bekshaev, A.Y.; Hanson, S.G.; Zenkova, C.Y.; Mokhun, I.I.; Jun, Z. Structured Light: Ideas and Concepts. Front. Phys.
**2020**, 8, 114. [Google Scholar] [CrossRef] - Forbes, A.; de Oliveira, M.; Dennis, M.R. Structured Light. Nat. Photonics
**2021**, 15, 253–262. [Google Scholar] [CrossRef] - Andrews, D.L. Structured Light and Its Applications: An Introduction to Phase Structured Beams and Nanoscale Optical Forces; Academic Press: Cambridge, MA, USA, 2011. [Google Scholar]
- Padgett, M.J.; Bowman, R. Tweezers with a Twist. Nat. Photonics
**2011**, 5, 343–348. [Google Scholar] [CrossRef] - Litchinitser, N.M. Structured Light Meets Structured Matter. Science
**2012**, 337, 1054–1055. [Google Scholar] [CrossRef] - Du, J.; Wang, J. High-Dimensional Structured Light Coding/Decoding for Freespace Optical Communications Free of Obstructions. Opt. Lett.
**2015**, 40, 4827–4857. [Google Scholar] [CrossRef] [PubMed] - Rosales-Guzmán, C.; Ndagano, B.; Forbes, A. A Review of Complex Vector Light Fields and Their Applications. J. Opt.
**2018**, 20, 123001. [Google Scholar] [CrossRef] - Angelo, J.P.; Chen, S.-J.; Ochoa, M.; Sunar, U.; Gioux, S.; Intes, X. Review of Structured Light in Diffuse Optical Imaging. J. Biomed. Opt.
**2018**, 24, 071602. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N. Vortex Beams with High-Order Cylindrical Polarization: Features of Focal Distributions. Appl. Phys. B
**2019**, 125, 100. [Google Scholar] [CrossRef] - Flamm, D.; Grossmann, D.G.; Sailer, M.; Kaiser, M.; Zimmermann, F.; Chen, K.; Jenne, M.; Kleiner, J.; Hellstern, J.; Tillkorn, C.; et al. Structured Light for Ultrafast Laser Micro- and Nanoprocessing. Opt. Eng.
**2021**, 60, 025105. [Google Scholar] [CrossRef] - Yang, Y.; Ren, Y.X.; Chen, M.; Arita, Y.; Rosales-Guzmán, C. Optical Trapping with Structured Light: A Review. Adv. Photon.
**2021**, 3, 034001. [Google Scholar] [CrossRef] - Porfirev, A.P.; Kuchmizhak, A.A.; Gurbatov, S.O.; Juodkazis, S.; Khonina, S.N.; Kulchin, Y.N. Phase Singularities and Optical Vortices in Photonics. Phys. Usp.
**2022**, 65, 789–811. [Google Scholar] [CrossRef] - Khonina, S.N.; Porfirev, A.P. Harnessing of Inhomogeneously Polarized Hermite–Gaussian Vector Beams to Manage the 3D Spin Angular Momentum Density Distribution. Nanophotonics
**2022**, 11, 697–712. [Google Scholar] [CrossRef] - Porfirev, A.; Khonina, S.; Kuchmizhak, A. Light–Matter Interaction Empowered by Orbital Angular Momentum: Control of Matter at The Micro-And Nanoscale. Prog. Quantum Electron.
**2023**, 88, 100459. [Google Scholar] [CrossRef] - Prasciolu, M.; Tamburini, F.; Anzolin, G.; Mari, E.; Melli, M.; Carpentiero, A.; Barbieri, C.; Romanato, F. Fabrication of a Three-Dimensional Optical Vortices Phase Mask for Astronomy by Means of Electron-Beam Lithography. Microelectron Eng.
**2009**, 86, 1103–1106. [Google Scholar] [CrossRef] - Massari, M.; Ruffato, G.; Gintoli, M.; Ricci, F.; Romanato, F. Fabrication and Characterization of High-Quality Spiral Phase Plates for Optical Applications. Appl. Opt.
**2015**, 54, 4077–4083. [Google Scholar] [CrossRef] - Forbes, A.; Dudley, A.; McLaren, M. Creation and Detection of Optical Modes with Spatial Light Modulators. Adv. Opt. Photon.
**2016**, 8, 200–227. [Google Scholar] [CrossRef] - Yue, F.; Wen, D.; Xin, J. Vector Vortex Beam Generation with a Single Plasmonic Metasurface. ACS Photonics
**2016**, 3, 1558. [Google Scholar] [CrossRef] - Nivas, J.J.J.; Allahyari, E.; Cardano, F.; Rubano, A.; Fittipaldi, R.; Vecchione, A.; Paparo, D.; Marrucci, L.; Bruzzese, R.; Amoruso, S. Surface Structures with Unconventional Patterns and Shapes Generated by Femtosecond Structured Light Fields. Sci. Rep.
**2018**, 8, 13613. [Google Scholar] [CrossRef] - Khonina, S.N.; Porfirev, A.P.; Kazanskiy, N.L. Variable Transformation of Singular Cylindrical Vector Beams Using Anisotropic Crystals. Sci. Rep.
**2020**, 10, 5590. [Google Scholar] [CrossRef] - Wang, J.; Liang, Y. Generation and Detection of Structured Light: A Review. Front. Phys.
**2021**, 9, 688284. [Google Scholar] [CrossRef] - Khonina, S.N.; Degtyarev, S.A.; Ustinov, A.V.; Porfirev, A.P. Metalenses for the Generation of Vector Lissajous Beams with a Complex Poynting Vector Density. Opt. Express
**2021**, 29, 18651–18662. [Google Scholar] [CrossRef] [PubMed] - Ahmed, H.; Kim, H.; Zhang, Y.; Intaravanne, Y.; Jang, J.; Rho, J.; Chen, S.; Chen, X. Optical Metasurfaces for Generating and Manipulating Optical Vortex Beams. Nanophotonics
**2022**, 11, 941–956. [Google Scholar] [CrossRef] - Zheng, S.; Zhao, Z.; Zhang, W. Versatile Generation and Manipulation of Phase-Structured Light Beams Using On-Chip Subwavelength Holographic Surface Gratings. Nanophotonics
**2023**, 12, 55–70. [Google Scholar] [CrossRef] - Zhu, L.; Wang, J. Simultaneous Generation of Multiple Orbital Angular Momentum (OAM) Modes Using a Single Phase-Only Element. Opt Express
**2015**, 23, 26221–26233. [Google Scholar] [CrossRef] - Mingyang, S.; Junmin, L.; Yanliang, H.; Shuqing, C.; Ying, L. Optical Orbital Angular Momentum Demultiplexing and Channel Equalization by Using Equalizing Dammann Vortex Grating. Adv. Condens. Matter Phys.
**2017**, 2017, 6293910. [Google Scholar] - Khonina, S.N.; Karpeev, S.V.; Paranin, V.D. A Technique for Simultaneous Detection of Individual Vortex States of Laguerre–Gaussian Beams Transmitted through an Aqueous Suspension of Microparticles. Opt. Lasers Eng.
**2018**, 105, 68–74. [Google Scholar] [CrossRef] - Qiao, Z.; Wan, Z.Y.; Xie, G.Q.; Wang, J.; Qian, L.J.; Fan, D.Y. Multi-Vortex Laser Enabling Spatial and Temporal Encoding. PhotoniX
**2020**, 1, 13. [Google Scholar] [CrossRef] - Ni, J.; Huang, C.; Zhou, L.M.; Gu, M.; Song, Q.; Kivshar, Y.; Qiu, C.W. Multidimensional Phase Singularities in Nanophotonics. Science
**2021**, 374, eabj0039. [Google Scholar] [CrossRef] - Maurer, C.; Jesacher, A.; Fürhapter, S.; Bernet, S.; Ritsch-Marte, M. Tailoring of Arbitrary Optical Vector Beams. New J. Phys.
**2007**, 9, 78. [Google Scholar] [CrossRef] - Man, Z.; Min, C.; Zhang, Y.; Shen, Z.; Yuan, X.-C. Arbitrary Vector Beams with Selective Polarization States Patterned by Tailored Polarizing Films. Laser Phys.
**2013**, 23, 105001. [Google Scholar] [CrossRef] - Chen, S.; Zhou, X.; Liu, Y.; Ling, X.; Luo, H.; Wen, S. Generation of Arbitrary Cylindrical Vector Beams on the Higher Order Poincaré Sphere. Opt. Lett.
**2014**, 39, 5274–5276. [Google Scholar] [CrossRef] [PubMed] - Millione, G.; Nguyen, T.A.; Leach, J.; Nolan, D.A.; Alfano, R.R. Using the Nonseparability of Vector Beams to Encode Information for Optical Communication. Opt. Lett.
**2015**, 40, 4887–4890. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Ustinov, A.V.; Fomchenkov, S.A.; Porfirev, A.P. Formation of Hybrid Higher-Order Cylindrical Vector Beams Using Binary Multi-Sector Phase Plates. Sci. Rep.
**2018**, 8, 14320. [Google Scholar] [CrossRef] - Wu, H.J.; Zhao, B.; Rosales-Guzmán, C.; Gao, W.; Shi, B.S.; Zhu, Z.H. Spatial-Polarization-Independent Parametric Up-Conversion of Vectorially Structured Light. Phys. Rev. Appl.
**2020**, 13, 064041. [Google Scholar] [CrossRef] - Huang, H.; Xie, G.; Yan, Y.; Ahmed, N.; Ren, Y.; Yue, Y.; Rogawski, D.; Willner, M.J.; Erkmen, B.I.; Birnbaum, K.M.; et al. 100 Tbit/s Free-Space Data Link Enabled by Three-Dimensional Multiplexing of Orbital Angular Momentum, Polarization, And Wavelength. Opt. Lett.
**2014**, 39, 197–200. [Google Scholar] [CrossRef] - Mitchell, K.J.; Turtaev, S.; Padgett, M.J.; Cižmár, T.; Phillips, D.B. High-Speed Spatial Control of the Intensity, Phase and Polarisation of Vector Beams Using a Digital Micro-Mirror Device. Opt Express
**2016**, 24, 29269–29282. [Google Scholar] [CrossRef] - Chen, Y.; Lin, Z.; Villers, S.B.D.; Rusch, L.A.; Shi, W. WDM-Compatible Polarization-Diverse OAM Generator and Multiplexer in Silicon Photonics. IEEE J. Sel. Top. Quantum Electron.
**2020**, 26, 6100107. [Google Scholar] [CrossRef] - Khonina, S.N.; Karpeev, S.V.; Porfirev, A.P. Sector Sandwich Structure: An Easy-to-Manufacture Way Towards Complex Vector Beam Generation. Opt. Express
**2020**, 28, 27628–27643. [Google Scholar] [CrossRef] - Dorrah, A.H.; Rubin, N.A.; Tamagnone, M.; Zaidi, A.; Capasso, F. Structuring Total Angular Momentum of Light Along the Propagation Direction with Polarization-Controlled Meta-Optics. Nat. Commun.
**2021**, 12, 6249. [Google Scholar] [CrossRef] - Khonina, S.N.; Kazanskiy, N.L.; Butt, M.A.; Karpeev, S.V. Optical Multiplexing Techniques and Their Marriage for On-Chip and Optical Fiber Communication: A Review. Opto-Electronic Adv.
**2022**, 5, 210127. [Google Scholar] - He, C.; Shen, Y.; Forbes, A. Towards Higher-Dimensional Structured Light. Light Sci. Appl.
**2022**, 11, 205. [Google Scholar] [CrossRef] - Dorn, R.; Quabis, S.; Leuchs, G. Sharper Focus for a Radially Polarized Light Beam. Phys. Rev. Lett.
**2003**, 91, 233901. [Google Scholar] [CrossRef] - Wang, H.; Shi, L.; Lukyanchuk, D.; Sheppard, C.; Chong, C.T. Creation of a Needle of Longitudinally Polarized Light in Vacuum Using Binary Optics. Nat. Photonics
**2008**, 2, 501–505. [Google Scholar] [CrossRef] - Jin, Y.; Allegre, O.J.; Perrie, W.; Abrams, K.; Ouyang, J.; Fearon, E.; Edwardson, S.P.; Dearden, G. Dynamic Modulation of Spatially Structured Polarization Fields for Real-Time Control of Ultrafast Laser-Material Interactions. Opt. Express
**2013**, 21, 25333–25343. [Google Scholar] [CrossRef] - Anoop, K.K.; Rubano, A.; Fittipaldi, R.; Wang, X.; Paparo, D.; Vecchione, A.; Marrucci, L.; Bruzzese, R.; Amoruso, S. Femtosecond Laser Surface Structuring of Silicon Using Optical Vortex Beams Generated by a Q-Plate. Appl. Phys. Lett.
**2014**, 104, 241604. [Google Scholar] [CrossRef] - Nivas, J.J.J.; Allahyari, E.; Amoruso, S. Direct Femtosecond Laser Surface Structuring with Complex Light Beams Generated by Q-Plates. Adv. Opt. Technol.
**2020**, 9, 53. [Google Scholar] [CrossRef] - Porfirev, A.; Khonina, S.; Ivliev, N.; Meshalkin, A.; Achimova, E.; Forbes, A. Writing and Reading with the Longitudinal Component of Light Using Carbazole-Containing Azopolymer Thin Films. Sci. Rep.
**2022**, 12, 3477. [Google Scholar] [CrossRef] - Porfirev, A.P.; Khonina, S.N.; Ivliev, N.A.; Fomchenkov, S.A.; Porfirev, D.P.; Karpeev, S.V. Polarization-Sensitive Patterning of Azopolymer Thin Films Using Multiple Structured Laser Beams. Sensors
**2023**, 23, 112. [Google Scholar] [CrossRef] - Moreno, I.; Davis, J.A.; Ruiz, I.; Cottrell, D.M. Decomposition of Radially and Azimuthally Polarized Beams Using a Circular-Polarization and Vortex-Sensing Diffraction Grating. Opt. Express
**2010**, 18, 7173–7183. [Google Scholar] [CrossRef] - Zhou, Z.-H.; Guo, Y.-K.; Zhu, L.-Q. Tight Focusing of Axially Symmetric Polarized Vortex Beams. Chin. Phys. B
**2014**, 23, 044201. [Google Scholar] [CrossRef] - Porfirev, A.P.; Ustinov, A.V.; Khonina, S.N. Polarization Conversion When Focusing Cylindrically Polarized Vortex Beams. Sci. Rep.
**2016**, 6, 6. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Porfirev, A.P.; Karpeev, S.V. Recognition of Polarization and Phase States of Light Based on the Interaction of Nonuniformly Polarized Laser Beams with Singular Phase Structures. Opt. Express
**2019**, 27, 18484–18492. [Google Scholar] [CrossRef] - Wang, X.-L.; Li, Y.; Chen, J.; Guo, C.-S.; Ding, J.; Wang, H.-T. A New Type of Vector Fields with Hybrid States of Polarization. Opt. Express
**2010**, 18, 10786–10795. [Google Scholar] [CrossRef] [PubMed] - Pu, J.; Zhang, Z. Tight Focusing of Spirally Polarized Vortex Beams. Opt. Laser Technol.
**2010**, 42, 186–191. [Google Scholar] [CrossRef] - Man, Z.; Min, C.; Zhu, S.; Yuan, X.-C. Tight Focusing of Quasi-Cylindrically Polarized Beams. J. Opt. Soc. Am. A
**2014**, 31, 373–378. [Google Scholar] [CrossRef] - Pan, Y.; Gao, X.-Z.; Zhang, G.-L.; Li, Y.; Tu, C.; Wang, H.-T. Spin Angular Momentum Density and Transverse Energy Flow of Tightly Focused Kaleidoscope-Structured Vector Optical Fields. APL Photonics
**2019**, 4, 096102. [Google Scholar] [CrossRef] - Zhao, Y.; Edgar, J.S.; Jeffries, G.D.M.; McGloin, D.; Chiu, D.T. Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam. Phys. Rev. Lett.
**2007**, 99, 073901. [Google Scholar] [CrossRef] - Zhu, J.; Chen, Y.; Zhang, Y.; Cai, X.; Yu, S. Spin and Orbital Angular Momentum and Their Conversion in Cylindrical Vector Vortices. Opt. Lett.
**2014**, 39, 4435–4438. [Google Scholar] [CrossRef] - Bliokh, K.Y.; Rodriguez-Fortuno, F.; Nori, F.; Zayats, A.V. Spin-Orbit Interactions of Light. Nat. Photonics
**2015**, 9, 796–808. [Google Scholar] [CrossRef] - Devlin, R.C.; Ambrosio, A.; Rubin, N.A.; Mueller, J.P.B.; Capasso, F. Arbitrary Spin-to–Orbital Angular Momentum Conversion of Light. Science
**2017**, 358, 896–901. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Golub, I. Vectorial Spin Hall Effect of Light Upon Tight Focusing. Opt. Lett.
**2022**, 47, 2166. [Google Scholar] [CrossRef] [PubMed] - Cheng, J.; Zhang, Z.; Mei, W.; Cao, Y.; Ling, X.; Chen, Y. Symmetry-Breaking Enabled Topological Phase Transitions in Spin-Orbit Optics. Opt. Express
**2023**, 31, 23621. [Google Scholar] [CrossRef] [PubMed] - Freund, I.; Soskin, M.S.; Mokhun, A.I. Elliptic Critical Points in Paraxial Optical Fields. Opt. Commun.
**2002**, 208, 223–253. [Google Scholar] [CrossRef] - Vyas, S.; Kozawa, Y.; Sato, S. Polarization Singularities in Superposition of Vector Beams. Opt. Express
**2013**, 21, 8972–8986. [Google Scholar] [CrossRef] - Kumar, V.; Philip, G.M.; Viswanathan, N.K. Formation and Morphological Transformation of Polarization Singularities: Hunting the Monstar. J. Opt.
**2013**, 15, 044027. [Google Scholar] [CrossRef] - Senthilkumaran, R.P.; Pal, S.K. Phase Singularities to Polarization Singularities. Int. J. Opt.
**2020**, 2020, 2812803. [Google Scholar] - Kotlyar, V.V.; Stafeev, S.S.; Nalimov, A.G. Sharp Focusing of a Hybrid Vector Beam with a Polarization Singularity. Photonics
**2021**, 8, 227. [Google Scholar] [CrossRef] - Bonse, J.; Höhm, S.; Kirner, S.V.; Rosenfeld, A.; Krüger, J. Laser-Induced Periodic Surface Structures—A Scientific Evergreen. IEEE J. Sel. Top. Quantum Electron.
**2016**, 23, 9000615. [Google Scholar] [CrossRef] - Richards, B.; Wolf, E. Electromagnetic Diffraction in Optical Sysems. II. Structure of the Aplanatic System. Proc. R. Soc. Lond.
**1959**, 253, 358. [Google Scholar] - Pereira, S.F.; van de Nes, A.S. Superresolution by Means of Polarisation, Phase and Amplitude Pupil Masks. Opt. Commun.
**2004**, 234, 119. [Google Scholar] [CrossRef] - Khonina, S.N.; Golub, I. Engineering the Smallest 3D Symmetrical Bright and Dark Focal Spots. J. Opt. Soc. Am. A
**2013**, 30, 2029–2033. [Google Scholar] [CrossRef] [PubMed] - Bliokh, K.Y.; Bekshaev, A.Y.; Nori, F. Extraordinary Momentum and Spin in Evanescent Waves. Nat. Commun.
**2014**, 5, 3300. [Google Scholar] [CrossRef] - Xu, X.; Nieto-Vesperinas, M. Azimuthal Imaginary Poynting Momentum Density. Phys. Rev. Lett.
**2019**, 123, 233902. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Porfirev, A.P.; Ustinov, A.V.; Kirilenko, M.S.; Kazanskiy, N.L. Tailoring of Inverse Energy Flow Profiles with Vector Lissajous Beams. Photonics
**2022**, 9, 121. [Google Scholar] [CrossRef] - Sheppard, C.J.R.; Choudhury, A. Annular Pupils, Radial Polarization, and Superresolution. Appl. Opt.
**2004**, 43, 4322–4327. [Google Scholar] [CrossRef] - Khonina, S.N.; Ustinov, A.V. Increased Reverse Energy Flux Area When Focusing a Linearly Polarized Annular Beam with Binary Plates. Opt. Lett.
**2019**, 44, 2008–2011. [Google Scholar] [CrossRef] - Man, Z.; Bai, Z.; Zhang, S.; Li, X.; Li, J.; Ge, X.; Zhang, Y.; Fu, S. Redistributing the energy flow of a tightly focused radially polarized optical field by designing phase masks. Opt. Express
**2018**, 26, 23935–23944. [Google Scholar] [CrossRef] - Man, Z.; Li, X.; Zhang, S.; Bai, Z.; Lyu, Y.; Li, J.; Ge, X.; Sun, Y.; Fu, S. Manipulation of the transverse energy flow of azimuthally polarized beam in tight focusing system. Opt. Commun.
**2019**, 431, 174–180. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Kovalev, A.A.; Nalimov, A.G. Energy density and energy flux in the focus of an optical vortex: Reverse flux of light energy. Opt. Lett.
**2018**, 43, 2921. [Google Scholar] [CrossRef]

**Figure 2.**Vector Debye theory of focusing an optical beam through a focusing system with a focal length f and maximum azimuthal angle Θ.

**Figure 3.**Formation of fields with non-uniform elliptical polarization with α(φ) = φ: field type at the input (

**a**,

**c**) and in the focal plane (

**b**,

**d**) for A

_{x}= A

_{y}(

**a**,

**b**) and for A

_{x}= 2A

_{y}(

**c**,

**d**).

**Figure 4.**Formation of fields with non-uniform elliptical polarization at α(φ) = 2φ. The arrows show local polarization directions.

**Figure 5.**Formation of fields with non-uniform elliptical polarization at α(φ) = 3φ. The arrows show local polarization directions.

**Figure 6.**Formation of fields with non-uniform elliptical polarization at α(φ) = 1.5φ. The arrows show local polarization directions.

**Figure 7.**Distribution of the longitudinal component ${S}_{z}$ (two upper lines: blue color corresponds to positive values, turquoise color is for negative values) and square of the real parts of the transverse components ${\left[\mathrm{Re}\left({S}_{x}\right)\right]}^{2}+{\left[\mathrm{Re}\left({S}_{y}\right)\right]}^{2}$ (bottom line: red color corresponds to x-component, green color corresponds to y-component, the direction of the transverse energy flow is shown by the black arrows) of the Poynting vector for various α(φ).

**Figure 8.**Formation of fields with non-uniform elliptical polarization with a simple trigonometric dependence β(φ). The arrows show local polarization directions.

**Figure 9.**Formation of fields with non-uniform elliptical polarization with a multiple trigonometric dependence β(φ). The arrows show local polarization directions.

**Figure 10.**Formation of fields with non-uniform elliptical polarization with a power trigonometric dependence β(φ). The arrows show local polarization directions.

**Figure 11.**Distribution of the longitudinal component ${S}_{z}$ (two upper lines) and square of the real parts of the transverse components ${\left[\mathrm{Re}\left({S}_{x}\right)\right]}^{2}+{\left[\mathrm{Re}\left({S}_{y}\right)\right]}^{2}$ (the bottom line) of the Poynting vector for various β(φ). The direction of the transverse energy flow is shown by the black arrows.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Khonina, S.N.; Ustinov, A.V.; Porfirev, A.P.
Generation of Light Fields with Controlled Non-Uniform Elliptical Polarization When Focusing on Structured Laser Beams. *Photonics* **2023**, *10*, 1112.
https://doi.org/10.3390/photonics10101112

**AMA Style**

Khonina SN, Ustinov AV, Porfirev AP.
Generation of Light Fields with Controlled Non-Uniform Elliptical Polarization When Focusing on Structured Laser Beams. *Photonics*. 2023; 10(10):1112.
https://doi.org/10.3390/photonics10101112

**Chicago/Turabian Style**

Khonina, Svetlana N., Andrey V. Ustinov, and Alexey P. Porfirev.
2023. "Generation of Light Fields with Controlled Non-Uniform Elliptical Polarization When Focusing on Structured Laser Beams" *Photonics* 10, no. 10: 1112.
https://doi.org/10.3390/photonics10101112