# Simplifying the Experimental Detection of the Vortex Topological Charge Based on the Simultaneous Astigmatic Transformation of Several Types and Levels in the Same Focal Plane

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Theoretical Foundations

_{2,±2}and defocus Z

_{2,0}(corresponding to a shift from the focus plane). Thus, the use of cylindrical lenses and other types of astigmatic transformations could lead to the need to select the best detection distance. However, this action can be replaced by changing the aberration magnitude (level).

#### 2.2. Simulation of Astigmatic Transformations

_{0}.

_{0}= 300 mm.

## 3. Proposed Approach Based on Multi-Channel DOEs

#### 3.1. Principle of Operation

_{1}, z

_{2}, … and z

_{j}) is shown in the upper part of Figure 1. A thorough description of the classical methods can be found in the according works [34,35,36,37,38,39,40,41]. The proposed approach based on a multi-channel DOE matched with astigmatic aberrations of different levels (implemented using a spatial light modulator (SLM)) and the detection of intensity distribution in a single focal plane (z

_{0}) is shown in the lower part of Figure 1. A detailed description of our method (step by step along the optical beam’s propagation) is given in Section 4.

#### 3.2. Simulation Results for Multi-Channel DOEs

_{0}is the number of DOE channels corresponding to astigmatic types of aberrations ${Z}_{n,2}(r,\mathsf{\phi})$, K

_{0}is the number of filter channels corresponding to different levels of aberrations ${\mathsf{\alpha}}_{j}$, and ${a}_{jN},{b}_{jN}$ are spatial carrier parameters along the X and Y axes.

_{0}= 5 different astigmatic aberrations ${Z}_{n,2}(r,\mathsf{\phi})$ (n = 2, 4, 6, 8, 10) with K

_{0}= 5 different levels ${\mathsf{\alpha}}_{j}$(${\mathsf{\alpha}}_{1}=0.1\lambda $, ${\mathsf{\alpha}}_{2}=0.25\lambda $, ${\mathsf{\alpha}}_{3}=0.5\lambda $, ${\mathsf{\alpha}}_{4}=0.75\lambda $, ${\mathsf{\alpha}}_{5}=\lambda $).

_{1}, z

_{2}, … and z

_{j}) in the classical scheme with an astigmatic or tilted lens. Thus, to select a good level of astigmatic transformation, it was not required to move the camera. It was only necessary to select suitable diffraction orders (marked with a frame), which allowed one to clearly determine the TC from the intensity pattern.

_{0}= 1) with a large range of α from 0.1λ to 5λ. The simulation results are shown in Table 6. The diffraction orders with astigmatic PSFs, which allowed for it to be possible to clearly determine the TC, were marked with a frame. We could confirm the following trend: with an increase in the TC of the vortex beam, an increase in the level of astigmatism α was required for correct detection. In addition, it required a higher level of α to form a well-defined astigmatism pattern when using the classical type of astigmatism (2, 2) (first column of Table 6). The astigmatism of type (6, 2) (third column of Table 6) was not suitable for vortex beams with a large TC due to the appearance of redundant artifacts in the intensity pattern.

#### 3.3. Optimization for High-Order TC

_{2,±2}and defocus Z

_{2,0}. In our approach, we proposed to replace defocusing with different levels of astigmatic aberration α. We carried out numerical studies at high TCs with various types of aberrations to determine the most appropriate transformation. The simulation results for the TC value l = 14 are shown in Table 7 and in Figure 4.

## 4. Laboratory Experiments

_{1}= 350 mm, f

_{2}= 300 mm, f

_{3}= 200 mm and f

_{4}= 250 mm); SLM1 was a transparent spatial light modulator (HOLOEYE LC 2012) for a vortex beam generation; SLM2 was a reflective spatial light modulator (HOLOEYE PLUTO VIS) for a multi-channel DOE implementation; D1 and D2 were circular apertures; M1 and M2 were mirrors; CAM was a video camera (ToupCam UCMOS08000KPB).

_{4}= 250 mm), which focused it onto the video camera (ToupCam UCMOS08000KPB CAM camera with a resolution of 3264 × 2448 and a pixel size of 1.67 µm).

## 5. Discussion

_{n,±}

_{2}type and their combination.

## 6. Conclusions

_{n,±}

_{2}type and their combination with defocusing aberrations, which was confirmed with the analytical representation of these aberrations as a super-position of Zernike functions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Nye, J.F.; Berry, M.V. Dislocations in Wave Trains. Proc. R. Soc. A
**1974**, 336, 165. [Google Scholar] - Bazhenov, V.Y.; Soskin, M.S.; Vasnetsov, M.V. Screw Dislocations in Light Wavefronts. J. Mod. Opt.
**1992**, 39, 985. [Google Scholar] [CrossRef] - Khonina, S.N.; Kotlyar, V.V.; Shinkarev, M.V.; Soifer, V.A.; Uspleniev, G.V. The Rotor Phase Filter. J. Mod. Opt.
**1992**, 39, 1147. [Google Scholar] [CrossRef] - Berry, M.V. Optical Vortices Evolving from Helicoidal Integer and Fractional Phase Steps. J. Opt. A Pure Appl. Opt.
**2004**, 6, 259. [Google Scholar] [CrossRef] - Shen, Y.; Wang, X.; Xie, Z.; Min, C.; Fu, X.; Liu, Q.; Gong, M.; Yuan, X. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci. Appl.
**2019**, 8, 90. [Google Scholar] [CrossRef] [PubMed] - Porfirev, A.P.; Kuchmizhak, A.A.; Gurbatov, S.O.; Juodkazis, S.; Khonina, S.N.; Kul’chin, Y.N. Phase singularities and optical vortices in photonics. Phys. Usp.
**2021**, 8. [Google Scholar] [CrossRef] - Allen, L.; Beijersbergen, M.W.; Spreeuw, R.J.C.; Woerdman, J.P. Orbital Angular Momentum of Light and the Transformation of Laguerre–Gaussian Laser Modes. Phys. Rev. A
**1992**, 45, 8185. [Google Scholar] [CrossRef] - Padgett, M.; Courtial, J.; Allen, L. Light’s Orbital Angular Momentum. Phys. Today
**2004**, 57, 35. [Google Scholar] [CrossRef] - Yao, A.M.; Padgett, M.J. Orbital Angular Momentum: Origins, Behavior and Applications. Adv. Opt. Photonics
**2011**, 3, 161. [Google Scholar] [CrossRef] - Padgett, M.J. Orbital Angular Momentum 25 Years on. Opt. Express
**2017**, 25, 11265. [Google Scholar] [CrossRef] - Fatkhiev, D.M.; Butt, M.A.; Grakhova, E.P.; Kutluyarov, R.V.; Stepanov, I.V.; Kazanskiy, N.L.; Khonina, S.N.; Lyubopytov, V.S.; Sultanov, A.K. Recent Advances in Generation and Detection of Orbital Angular Momentum Optical Beams—A Review. Sensors
**2021**, 21, 4988. [Google Scholar] [CrossRef] [PubMed] - Gibson, G.; Courtial, J.; Padgett, M.; Vasnetsov, M.; Pas’ko, V.; Barnett, S.; Franke-Arnold, S. Free-Space Information Transfer Using Light Beams Carrying Orbital Angular Momentum. Opt. Express
**2004**, 12, 5448. [Google Scholar] [CrossRef] [PubMed] - Wang, J.; Yang, J.-Y.; Fazal, I.M.; Ahmed, N.; Yan, Y.; Huang, H.; Ren, Y.; Yue, Y.; Dolinar, S.; Tur, M.; et al. Terabit Free-Space Data Transmission Employing Orbital Angular Momentum Multiplexing. Nat. Photonics
**2012**, 6, 488. [Google Scholar] [CrossRef] - Bozinovic, N.; Yue, Y.; Ren, Y.; Tur, M.; Kristensen, P.; Huang, H.; Willner, A.E.; Ramachandran, S. Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers. Science
**2013**, 340, 1545. [Google Scholar] [CrossRef] [PubMed] - Zhu, F.; Huang, S.; Shao, W.; Zhang, J.; Chen, M.; Zhang, W.; Zeng, J. Freespace Optical Communication Link using Perfect Vortex Beams Carrying Orbital Angular Momentum (OAM). Opt. Commun.
**2017**, 396, 50. [Google Scholar] [CrossRef] - Karpeev, S.V.; Podlipnov, V.V.; Ivliev, N.A.; Khonina, S.N. High-speed Format 1000BASESX/LX Transmission through the Atmosphere by Vortex Beams near IR Range with Help Modified SFP-Transivers DEM-310GT. Comput. Opt.
**2020**, 44, 578. [Google Scholar] [CrossRef] - Khonina, S.N.; Karpeev, S.V.; Butt, M.A. Spatial-Light-Modulator-Based Multichannel Data Transmission by Vortex Beams of Various Orders. Sensors
**2021**, 21, 2988. [Google Scholar] [CrossRef] - Grier, D.A. Revolution in optical manipulation. Nature
**2003**, 424, 14. [Google Scholar] [CrossRef] - Paez-Lopez, R.; Ruiz, U.; Arrizon, V.; Ramos-Garcia, R. Optical manipulation using optimal annular vortices. Opt. Lett.
**2016**, 41, 4138. [Google Scholar] [CrossRef] - Shi, L.; Lindwasser, L.; Wang, W.; Alfano, R.; Rodriguez-Contreras, A. Propagation of Gaussian and Laguerre-Gaussian vortex beams through mouse brain tissue. J. Biophotonics
**2007**, 10, 1756. [Google Scholar] [CrossRef] - Sirenko, A.A.; Marsik, P.; Bernhard, C.; Stanislavchuk, T.N.; Kiryukhin, V.; Cheong, W. Terahertz Vortex Beam as a Spectroscopic Probe of Magnetic Excitations. Phys. Rev. Lett.
**2019**, 122, 237401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Khonina, S.N.; Kotlyar, V.; Soifer, V.; Paakkonen, P.; Simonen, J.; Turunen, J. An Analysis of the Angular Momentum of a Light Field in Terms of Angular Harmonics. J. Mod. Opt.
**2001**, 48, 1543. [Google Scholar] [CrossRef] - Moreno, I.; Davis, J.A.; Pascoguin, B.L.; Mitry, M.J.; Cottrell, D.M. Vortex Sensing Diffraction Gratings. Opt. Lett.
**2009**, 34, 2927. [Google Scholar] [CrossRef] [PubMed] - Fu, S.; Zhang, S.; Wang, T.; Gao, C. Measurement of Orbital Angular Momentum Spectra of Multiplexing Optical Vortices. Opt. Express
**2016**, 24, 6240. [Google Scholar] [CrossRef] - D’Errico, A.; D’Amelio, R.; Piccirillo, B.; Cardano, F.; Marrucci, L. Measuring the Complex Orbital Angular Momentum Spectrum and Spatial Mode Decomposition of Structured Light Beams. Optica
**2017**, 4, 1350. [Google Scholar] [CrossRef] - Fu, S.; Zhai, Y.; Wang, T.; Yin, C.; Gao, C. Orbital Angular Momentum Channel Monitoring of Coaxially Multiplexed Vortices by Diffraction Pattern Analysis. Appl. Opt.
**2018**, 57, 1056. [Google Scholar] [CrossRef] - Berkhout, G.C.G.; Lavery, M.P.J.; Courtial, J.; Beijersbergen, M.W.; Padgett, M.J. Efficient Sorting of Orbital Angular Momentum States of Light. Phys. Rev. Lett.
**2010**, 105, 153601. [Google Scholar] [CrossRef] - Mirhosseini, M.; Malik, M.; Shi, Z.; Boyd, R.W. Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun.
**2013**, 4, 2781. [Google Scholar] [CrossRef] - Wen, Y.; Chremmos, I.; Chen, Y.; Zhu, J.; Zhang, Y.; Yu, S. Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes. Phys. Rev. Lett.
**2018**, 120, 193904. [Google Scholar] [CrossRef] - Wen, Y.; Chremmos, I.; Chen, Y.; Zhu, G.; Zhang, J.; Zhu, J.; Zhang, Y.; Liu, J.; Yu, S. Compact and High-Performance Vortex Mode Sorter for Multi-Dimensional Multiplexed Fiber Communication Systems. Optica
**2020**, 7, 254. [Google Scholar] [CrossRef] - Abramochkin, E.; Volostnikov, V. Beam Transformations and Nontransformed Beams. Opt. Commun.
**1991**, 83, 123. [Google Scholar] [CrossRef] - Beijersbergen, M.W.; Allen, L.; van der Veen, H.E.L.O.; Woerdman, J.P. Astigmatic Laser Mode Converters and Transfer of Orbital Angular Momentum. Opt. Commun.
**1993**, 96, 123. [Google Scholar] [CrossRef] - Khonina, S.N.; Kotlyar, V.V.; Soifer, V.A.; Jefimovs, K.; Paakkonen, P.; Turunen, J. Astigmatic Bessel laser beams. J. Mod. Opt.
**2004**, 51, 677. [Google Scholar] [CrossRef] - Bekshaev, A.Y.; Soskin, M.S.; Vasnetsov, M.V. Transformation of Higher-Order Optical Vortices upon Focusing by a Astigmatic Lens. Opt. Commun.
**2004**, 241, 237. [Google Scholar] [CrossRef] - Abramochkin, E.; Razueva, E.; Volostnikov, V. General Astigmatic Transform of Hermite-Laguerre-Gaussian Beams. J. Opt. Soc. Am. A
**2010**, 27, 2506. [Google Scholar] [CrossRef] [PubMed] - Reddy, S.G.; Prabhakar, S.; Aqadhi, A.; Banerji, J.; Singh, R.P. Propagation of an Arbitrary Vortex Pair through an Astigmatic Optical System and Determination of Its Topological Charge. J. Opt. Soc. Am. A
**2014**, 31, 1295. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Kovalev, A.A.; Porfirev, A.P. Determination of an Optical Vortex Topological Charge using an Astigmatic Transform. Comput. Opt.
**2016**, 40, 781. [Google Scholar] [CrossRef] - Porfirev, A.P.; Khonina, S.N. Astigmatic Transformation of Optical Vortex Beams with High-Order Cylindrical Polarization. J. Opt. Soc. Am. B
**2019**, 36, 2193. [Google Scholar] [CrossRef] - Vaity, P.; Banerji, J.; Singh, R.P. Measuring the Topological Charge of an Optical Vortex by Using a Tilted Convex Lens. Phys. Lett. A
**2013**, 377, 1154. [Google Scholar] [CrossRef] - Peng, Y.; Gan, X.; Ju, P.; Wang, Y.; Zhao, J. Measuring Topological Charges of Optical Vortices with Multi-Singularity using a Cylindrical Lens. Chin. Phys. Lett.
**2015**, 32, 024201. [Google Scholar] [CrossRef] - Liu, P.; Cao, Y.; Lu, Z.; Lin, G. Probing Arbitrary Laguerre–Gaussian Beams and Pairs through a Tilted Biconvex Lens. J. Opt.
**2021**, 23, 025002. [Google Scholar] [CrossRef] - Thaning, A.; Jaroszewicz, Z.; Friberg, A.T. Diffractive Axicons in Oblique Illumination: Analysis and Experiments and Comparison with Elliptical Axicons. Appl. Opt.
**2003**, 42, 9. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Kazanskiy, N.L.; Khorin, P.A.; Butt, M.A. Modern Types of Axicons: New Functions and Applications. Sensors
**2021**, 21, 6690. [Google Scholar] [CrossRef] - Dwivedi, R.; Sharma, P.; Jaiswal, V.K.; Mehrotra, R. Elliptically Squeezed Axicon Phase for Detecting Topological Charge of Vortex Beam. Opt. Commun.
**2021**, 485, 126710. [Google Scholar] [CrossRef] - Almazov, A.A.; Khonina, S.N.; Kotlyar, V.V. How the Tilt of a Phase Diffraction Optical Element Affects the Properties of Shaped Laser Beams Matched with a Basis of Angular Harmonics. J. Opt. Technol.
**2006**, 73, 633. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Kovalev, A.A.; Porfirev, A.P. Astigmatic Transforms of an Optical Vortex for Measurement of Its Topological Charge. Appl. Opt.
**2017**, 56, 4095. [Google Scholar] [CrossRef] [PubMed] - Hacyan, S.; Jáuregui, R. Evolution of Optical Phase and Polarization Vortices in Birefringent Media. J. Opt. A Pure Appl. Opt.
**2009**, 11, 085204. [Google Scholar] [CrossRef] - Zusin, D.H.; Maksimenka, R.; Filippov, V.V.; Chulkov, R.V.; Perdrix, M.; Gobert, O.; Grabtchikov, A.S. Bessel Beam Transformation by Anisotropic Crystals. J. Opt. Soc. Am. A
**2010**, 27, 1828. [Google Scholar] [CrossRef] - Khonina, S.N.; Paranin, V.D.; Ustinov, A.V.; Krasnov, A.P. Astigmatic Transformation of Bessel Beams in a Uniaxial Crystal. Opt. Appl.
**2016**, 46, 5. [Google Scholar] - 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] - Zheng, S.; Wang, J. Measuring Orbital Angular Momentum (OAM) States of Vortex Beams with Annular Gratings. Sci. Rep.
**2017**, 7, 40781. [Google Scholar] [CrossRef] [PubMed] - Rasouli, S.; Fathollazade, S.; Amiri, P. Simple, Efficient and Reliable Characterization of Laguerre-Gaussian Beams with Non-Zero Radial Indices in Diffraction from an Amplitude Parabolic-Line Linear Grating. Opt. Express
**2021**, 29, 29661. [Google Scholar] [CrossRef] [PubMed] - Amiri, P.; Mardan Dezfouli, A.; Rasouli., S. Efficient characterization of optical vortices via diffraction from parabolic-line linear gratings. J. Opt. Soc. Am. B
**2020**, 37, 2668. [Google Scholar] [CrossRef] - Rasouli, S.; Amiri, P.; Kotlyar, V.; Kovalev, A. Characterization of a Pair of Superposed Vortex Beams Having Different Winding Numbers via Diffraction from a Quadratic Curved-Line Grating. J. Opt. Am. B
**2021**, 38, 2267. [Google Scholar] [CrossRef] - Bekshaev, A.Y.; Karamoch, A.I. Astigmatic Telescopic Transformation of a High-Order Optical Vortex. Opt. Commun.
**2008**, 281, 5687. [Google Scholar] [CrossRef] - Porfirev, A.P.; Khonina, S.N. Experimental Investigation of Multi-Order Diffractive Optical Elements Matched with Two Types of Zernike Functions. Proc. SPIE
**2016**, 9807, 98070E. [Google Scholar] - Khonina, S.N.; Karpeev, S.V.; Porfirev, A.P. Wavefront Aberration Sensor Based on a Multichannel Diffractive Optical Element. Sensors
**2020**, 20, 3850. [Google Scholar] [CrossRef] - Khorin, P.A.; Khonina, S.N. Aberration-Matched Filters for Vortex Beams Transformations. Proc. SPIE
**2022**, 12295, 122950R. [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. [Google Scholar] [CrossRef] - Arnaud, J.; Kogelnik, H. Gaussian Beams with General Astigmatism. Appl. Opt.
**1969**, 25, 2908. [Google Scholar] [CrossRef] - Born, M.; Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed.; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Lakshminarayanana, V.; Fleck, A. Zernike Polynomials: A Guide. J. Mod. Opt.
**2011**, 58, 545. [Google Scholar] [CrossRef] - Vinogradova, M.B.; Rudenko, O.V.; Sukhorukov, A.P. Wave Theory, 2nd ed.; Nauka Publisher: Moscow, Russia, 1979. [Google Scholar]
- Abramochkin, E.; Losevsky, N.; Volostnikov, V. Generation of Spiral-Type Laser Beams. Opt. Commun.
**1997**, 141, 59. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Soifer, V.A.; Khonina, S.N. Rotation of Multimode Gauss-Laguerre Light Beams in Free Space. Tech. Phys. Lett.
**1997**, 23, 657. [Google Scholar] [CrossRef] - Khonina, S.N.; Koltyar, V.V.; Soifer, V.A. Techniques for encoding composite diffractive optical elements. Proc. SPIE
**2003**, 5036, 493–498. [Google Scholar] - Kogelnik, H.; Li, T. Laser Beams and Resonators. Appl. Opt.
**1966**, 5, 1550. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Khonina, S.N.; Soifer, V.A. Generalized Hermite Beams in Free Space. Optik
**1998**, 108, 20–26. [Google Scholar] - Kotlyar, V.V.; Kovalev, A.A.; Porfirev, A.P.; Kozlova, E.S. Three Different Types of Astigmatic Hermite-Gaussian Beams with Orbital Angular Momentum. J. Opt.
**2019**, 21, 115601. [Google Scholar] [CrossRef] - Khonina, S.N.; Balalayev, S.A.; Skidanov, R.V.; Kotlyar, V.V.; Päivänranta, B.; Turunen, J. Encoded binary diffractive element to form hyper-geometric laser beams. J. Opt.
**2009**, 11, 065702. [Google Scholar] [CrossRef] - Khonina, S.N.; Skidanov, R.V.; Kotlyar, V.V.; Jefimovs, K.; Turunen, J. Phase diffractive filter to analyze an output step-index fiber beam. Proc. SPIE Int. Soc. Opt. Eng.
**2004**, 5182, 251. [Google Scholar] - Guo, H.; Korablinova, N.; Ren, Q.; Bille, J. Wavefront Reconstruction with Artificial Neural Networks. Opt. Express
**2006**, 14, 6456. [Google Scholar] [CrossRef] - Nishizaki, Y.; Valdivia, M.; Horisaki, R.; Kitaguchi, K.; Saito, M.; Tanida, J.; Vera, E. Deep Learning Wavefront Sensing. Opt. Express
**2019**, 27, 240. [Google Scholar] [CrossRef] [PubMed] - Rodin, I.A.; Khonina, S.N.; Serafimovich, P.G.; Popov, S.B. Recognition of Wavefront Aberrations Types Corresponding to Single Zernike Functions from the Pattern of the Point Spread Function in the Focal Plane using Neural Networks. Comput. Opt.
**2020**, 44, 923. [Google Scholar] [CrossRef] - Khorin, P.A.; Dzyuba, A.P.; Serafimovich, P.G.; Khonina, S.N. Neural Networks Application to Determine the Types and Magnitude of Aberrations from the Pattern of the Point Spread Function out of the Focal Plane. J. Phys. Conf. Ser.
**2021**, 2086, 012148. [Google Scholar] [CrossRef] - Khonina, S.N.; Khorin, P.A.; Serafimovich, P.G.; Dzyuba, A.P.; Georgieva, A.O.; Petrov, N.V. Analysis of the Wavefront Aberrations based on Neural Networks Processing of the Interferograms with a Conical Reference Beam. Appl. Phys. B
**2022**, 128, 60. [Google Scholar] [CrossRef]

**Figure 1.**Principle of operation for determining the vortex TC (l = 3) in a standard way (upper part) using a tilted lens and detecting the intensity distribution in several planes (z

_{1}, z

_{2}, …, z

_{j}) and the proposed approach (lower part) based on multi-channel DOE matched with astigmatic aberrations $\mathrm{exp}\left[ik\mathsf{\alpha}{Z}_{n,2}(r,\mathsf{\phi})\right]$ of different levels α in a single focal plane z

_{0}.

**Figure 2.**Amplitude (

**a**) and phase (

**b**) of a 25-channel amplitude-phase DOE, (

**c**) the coded phase of DOE matched to different astigmatic aberrations ${Z}_{n,2}(r,\mathsf{\phi})$ (n = 2, 4, 6, 8, 10) with various levels ${\mathsf{\alpha}}_{j}$, intensity distribution in the focal plane (

**d**) at the Gaussian beam illumination (correspondence of aberrated PSF to diffraction orders is shown).

**Figure 3.**Detailed simulation results for multi-channel DOEs matched with one type of aberration ${Z}_{4,2}(r,\mathsf{\phi})$ with α ranging from 2.2λ to 3λ when illuminated by a vortex beam with TC l = 7.

**Figure 4.**Detailed simulation results for multi-channel DOE matched with one type of aberration ${Z}_{2,2}(r,\mathsf{\phi})$ and ${Z}_{4,2}(r,\mathsf{\phi})$ with α ranging from 4.2λ to 5λ when illuminated with a vortex beam with TC l = 14.

**Figure 5.**The experimental setup for detecting the TC of a vortex beam using a multi-channel DOE. Laser is a solid-state laser (λ = 532 nm); PH is a pinhole (hole size of 40 μm); L1, L2, L3 and L4 are spherical lenses (f

_{1}= 350 mm, f

_{2}= 300 mm, f

_{3}= 200 mm and f

_{4}= 250 mm); SLM1 is a transparent spatial light modulator (HOLOEYE LC 2012); SLM2 is a reflective spatial light modulator (HOLOEYE PLUTO VIS); D1 and D2 are circular apertures; M1 and M2 are mirrors; CAM is a ToupCam UCMOS08000KPB video camera.

**Figure 6.**The experimentally registered intensity distribution for the Gaussian beam illumination of the multi-channel DOE matched with different astigmatic aberrations of type (n, 2) of different levels of α from 0.1λ to λ.

n | m | Aberration Type | Mathematical Representation | Phase |
---|---|---|---|---|

2 | 2 | Astigmatism | $\sqrt{6}{r}^{2}\mathrm{cos}(2\mathsf{\phi})={c}_{1,1}({x}^{2}-{y}^{2})$ | |

4 | 2 | Fourth order astigmatism | $\sqrt{10}(4{r}^{4}-3{r}^{2})\mathrm{cos}(2\mathsf{\phi})={c}_{2,1}({x}^{4}-{y}^{4})-{c}_{2,2}({x}^{2}-{y}^{2})$ | |

6 | 2 | Sixth order astigmatism | $\begin{array}{l}\sqrt{14}(15{r}^{6}-20{r}^{4}+6{r}^{2})\mathrm{cos}(2\mathsf{\phi})=\\ {c}_{3,1}({x}^{4}-{y}^{4})({x}^{2}+{y}^{2})-{c}_{3,2}({x}^{4}-{y}^{4})+{c}_{3,3}({x}^{2}-{y}^{2})\end{array}$ | |

8 | 2 | Eighth order astigmatism | $\begin{array}{l}\sqrt{18}(56{r}^{8}-105{r}^{6}+60{r}^{4}-10{r}^{2})\mathrm{cos}(2\mathsf{\phi})=\\ {c}_{4,1}{x}^{8}+{c}_{4,2}{x}^{6}{y}^{2}-{c}_{4,3}{x}^{6}-{c}_{4,4}{x}^{4}{y}^{2}+{c}_{4,5}{x}^{4}-{c}_{4,6}{x}^{2}{y}^{6}+\\ +{c}_{4,7}{x}^{2}{y}^{4}-{c}_{4,8}{x}^{2}-{c}_{4,9}{y}^{8}+{c}_{4,10}{y}^{6}-{c}_{4,11}{y}^{4}+{c}_{4,12}{y}^{2}\end{array}$ | |

q | 2 | qth order astigmatism | $\sqrt{\frac{q+1}{\mathrm{\pi}}}{\displaystyle \sum _{s=0}^{q/2-1}\frac{{\left(-1\right)}^{p}\left(q-s\right)!}{s!\left(q/2+1-s\right)!\left(q/2-1-s\right)!}}{\left(\frac{r}{{r}_{0}}\right)}^{q-2s}\mathrm{cos}\left(2\mathsf{\phi}\right)$ |

**Table 2.**Correspondence of some types of astigmatic transformations of Equation (4) and Zernike functions.

Astigmatic Transformation Equation (4) | Mathematical Representation as Zernike Functions of Equation (5) | Phase |
---|---|---|

xy | $r\mathrm{cos}(\mathsf{\phi})\cdot r\mathrm{sin}(\mathsf{\phi})={d}_{1,1}{Z}_{2,-2}$ | |

${x}^{2}$ | ${r}^{2}{\mathrm{cos}}^{2}(\mathsf{\phi})={d}_{2,1}{Z}_{2,0}+{d}_{2,2}{Z}_{2,2}$ | |

${y}^{2}$ | ${r}^{2}{\mathrm{sin}}^{2}(\mathsf{\phi})={d}_{3,1}{Z}_{2,0}-{d}_{3,2}{Z}_{2,2}$ | |

${x}^{2}-{y}^{2}$ | ${r}^{2}{\mathrm{cos}}^{2}(\mathsf{\phi})-{r}^{2}{\mathrm{sin}}^{2}(\mathsf{\phi})={d}_{4,1}{Z}_{2,2}$ | |

${(x-y)}^{2}$ | ${r}^{2}{\mathrm{cos}}^{2}(\mathsf{\phi})-2{r}^{2}\mathrm{cos}(\mathsf{\phi})\mathrm{sin}(\mathsf{\phi})+{r}^{2}{\mathrm{sin}}^{2}(\mathsf{\phi})={d}_{5,1}{Z}_{2,0}-{d}_{5,2}{Z}_{2,-2}$ | |

${(x+y)}^{2}$ | ${r}^{2}{\mathrm{cos}}^{2}(\mathsf{\phi})+2{r}^{2}\mathrm{cos}(\mathsf{\phi})\mathrm{sin}(\mathsf{\phi})+{r}^{2}{\mathrm{sin}}^{2}(\mathsf{\phi})={d}_{6,1}{Z}_{2,0}+{d}_{6,2}{Z}_{2,-2}$ |

**Table 3.**Simulation results of astigmatic transformations of vortex beams ${\mathsf{\Psi}}_{0,l}(r,\mathsf{\phi})$ with aberrations of the form $\mathrm{exp}\left[ik\mathsf{\alpha}{Z}_{n,2}(r,\mathsf{\phi})\right]$ in the focal plane.

Astigmatic Parameters | Topological Charge | |||||
---|---|---|---|---|---|---|

l = −5 | l = −3 | l = −1 | l = 1 | l = 3 | l = 5 | |

n = 2, α = 3λ | ||||||

n = 4, α = λ | ||||||

n = 6, α = λ | ||||||

n = 8, α = λ |

**Table 4.**Simulation results of astigmatic transformation of vortex beams ${\mathsf{\Psi}}_{0,l}(r,\mathsf{\phi})$ with aberrations of the form $\mathrm{exp}\left[ik\mathsf{\alpha}{Z}_{n,2}(r,\mathsf{\phi})\right]$ at various distances Δz from the focal plane.

$\mathbf{Vortex}\text{}\mathbf{Beam}\text{}{\mathsf{\Psi}}_{0,\mathit{l}}(\mathit{r},\mathsf{\phi})$ | Astigmatic Parameters | Intensity Distributions at Various Distances Δz from the Focal Plane | |||
---|---|---|---|---|---|

0 mm | 100 mm | 200 mm | 300 mm | ||

TC l = 1 | n = 2, α = 3λ | ||||

n = 4, α = λ | |||||

n = 6, α = λ | |||||

TC l = 3 | n = 2, α = 3λ | ||||

n = 4, α = λ | |||||

n = 6, α = λ | |||||

TC l = 5 | n = 2, α = 3λ | ||||

n = 4, α = λ | |||||

n = 6, α = λ |

l = 1 | l = 3 | l = 5 |
---|---|---|

**Table 6.**Simulation results for multi-channel DOEs matched with one type of aberration ${Z}_{n,2}(r,\mathsf{\phi})$ (n = 2, 4, 6) with α ranging from 0.2λ to 5λ when illuminated with a vortex beam with a TC l = 3, 5, 7.

Vortex TCl= 3 | ||

n = 2 | n = 4 | n = 6 |

Vortex TCl = 5 | ||

n = 2 | n = 4 | n = 6 |

Vortex TCl = 7 | ||

n = 2 | n = 4 | n = 6 |

**Table 7.**Simulation results for multi-channel DOE matched with one type of aberration ${Z}_{n,2}(r,\mathsf{\phi})$ (n = 2, 4, 6) with α ranging from 0.2λ to 5λ when illuminated with a vortex beam with TC l = 14.

n = 2 | n = 4 | n = 6 |
---|---|---|

**Table 8.**The experimental results for the 25-channel DOE when illuminated with a vortex beam with TC l = 1, 2, 3, 5 (diffractive orders with astigmatic intensity pictures convenient for TC recognition were marked with frames).

TC | 25-Channel DOE Action | TC | 25-Channel DOE Action |
---|---|---|---|

l = 1 | l = 2 | ||

l = 3 | l = 5 |

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**

Khorin, P.A.; Khonina, S.N.; Porfirev, A.P.; Kazanskiy, N.L.
Simplifying the Experimental Detection of the Vortex Topological Charge Based on the Simultaneous Astigmatic Transformation of Several Types and Levels in the Same Focal Plane. *Sensors* **2022**, *22*, 7365.
https://doi.org/10.3390/s22197365

**AMA Style**

Khorin PA, Khonina SN, Porfirev AP, Kazanskiy NL.
Simplifying the Experimental Detection of the Vortex Topological Charge Based on the Simultaneous Astigmatic Transformation of Several Types and Levels in the Same Focal Plane. *Sensors*. 2022; 22(19):7365.
https://doi.org/10.3390/s22197365

**Chicago/Turabian Style**

Khorin, Pavel A., Svetlana N. Khonina, Alexey P. Porfirev, and Nikolay L. Kazanskiy.
2022. "Simplifying the Experimental Detection of the Vortex Topological Charge Based on the Simultaneous Astigmatic Transformation of Several Types and Levels in the Same Focal Plane" *Sensors* 22, no. 19: 7365.
https://doi.org/10.3390/s22197365