[1] Erhard M, Fickler R, Krenn M et al. Twisted photons: new quantum perspectives in high dimensions[J]. Light: Science & Applications, 7, 17146(2018).

[2] Tamburini F, Anzolin G, Umbriaco G et al. Overcoming the Rayleigh criterion limit with optical vortices[J]. Physical Review Letters, 97, 163903(2006).

[3] Zhou Z Y, Ding D S, Jiang Y K et al. Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals[J]. Optics Express, 22, 20298-20310(2014).

[4] Willner A E, Zhao Z, Liu C et al. Perspectives on advances in high-capacity, free-space communications using multiplexing of orbital-angular-momentum beams[J]. APL Photonics, 6, 030901(2021).

[5] Wang J, Liu J, Li S H et al. Orbital angular momentum and beyond in free-space optical communications[J]. Nanophotonics, 11, 645-680(2022).

[6] Liu J, Du Q, Liu F N et al. Vortex beam phase correction based on deep phase estimation network[J]. Acta Optica Sinica, 43, 0601013(2023).

[7] Ye Y E, Li J Y, Cao M et al. Accuracy recognition of orbital angular momentum of dual-mode vortex beams[J]. Laser & Optoelectronics Progress, 58, 1811021(2021).

[8] Li Y S, Chen J, Fu G K et al. Measurement of topological charge of obstructed wandering vortex beams[J]. Acta Optica Sinica, 43, 0226002(2023).

[9] Shen Y J, Wang X J, Xie Z W et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities[J]. Light: Science & Applications, 8, 90(2019).

[10] Bai Y H, Lü H R, Fu X et al. Vortex beam: generation and detection of orbital angular momentum[J]. Chinese Optics Letters, 20, 012601(2022).

[11] Li P Y, Wang B, Song X B et al. Non-destructive identification of twisted light[J]. Optics Letters, 41, 1574-1577(2016).

[12] Zhu J, Zhang P, Fu D Z et al. Probing the fractional topological charge of a vortex light beam by using dynamic angular double slits[J]. Photonics Research, 4, 187-190(2016).

[13] Berkhout G C G, Lavery M P J, Courtial J et al. Efficient sorting of orbital angular momentum states of light[J]. Physical Review Letters, 105, 153601(2010).

[14] Berger B, Kahlert M, Schmidt D et al. Spectroscopy of fractional orbital angular momentum states[J]. Optics Express, 26, 32248-32258(2018).

[15] Alperin S N, Niederriter R D, Gopinath J T et al. Quantitative measurement of the orbital angular momentum of light with a single, stationary lens[J]. Optics Letters, 41, 5019-5022(2016).

[16] Deng D, Lin M C, Li Y et al. Precision measurement of fractional orbital angular momentum[J]. Physical Review Applied, 12, 014048(2019).

[17] Zhang H, Zeng J, Lu X Y et al. Review on fractional vortex beam[J]. Nanophotonics, 11, 241-273(2022).

[18] Na Y B, Ko D K. Deep-learning-based high-resolution recognition of fractional-spatial-mode-encoded data for free-space optical communications[J]. Scientific Reports, 11, 2678(2021).

[19] Du H B, Chen J, Fu G K et al. Recognition of orbital angular momentum of fractional perfect optical vortex beam based on convolutional neural network and multiaperture interferometer[J]. Acta Optica Sinica, 43, 0426001(2023).

[20] Zhou H P, Pan Z Z, Dedo M I et al. High-efficiency and high-precision identification of transmitting orbital angular momentum modes in atmospheric turbulence based on an improved convolutional neural network[J]. Journal of Optics, 23, 065701(2021).

[21] Liu X L, Chen X D, Lin Z L et al. Deep learning aided measurement of topological charge number of strongly scattered vortex beams[J]. Acta Optica Sinica, 42, 1426001(2022).

[22] Li Z X, Li X, Jia H J et al. High-efficiency anti-interference OAM-FSO communication system based on Phase compression and improved CNN[J]. Optics Communications, 537, 129120(2023).

[23] Hao Y, Zhao L, Huang T et al. High-accuracy recognition of orbital angular momentum modes propagated in atmospheric turbulences based on deep learning[J]. IEEE Access, 8, 159542-159551(2020).

[24] Liu J M, Wang P P, Zhang X K et al. Deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication[J]. Optics Express, 27, 16671-16688(2019).

[25] Liu Z W, Yan S, Liu H G et al. Superhigh-resolution recognition of optical vortex modes assisted by a deep-learning method[J]. Physical Review Letters, 123, 183902(2019).

[26] Na Y B, Ko D K. Adaptive demodulation by deep-learning-based identification of fractional orbital angular momentum modes with structural distortion due to atmospheric turbulence[J]. Scientific Reports, 11, 23505(2021).

[27] Cao M, Yin Y L, Zhou J W et al. Machine learning based accurate recognition of fractional optical vortex modes in atmospheric environment[J]. Applied Physics Letters, 119, 141103(2021).

[28] Zhou J W, Yin Y L, Tang J H et al. Recognition of high-resolution optical vortex modes with deep residual learning[J]. Physical Review A, 106, 013519(2022).

[29] Berry M V. Optical vortices evolving from helicoidal integer and fractional phase steps[J]. Journal of Optics A: Pure and Applied Optics, 6, 259-268(2004).

[30] Lochab P, Senthilkumaran P, Khare K. Propagation of converging polarization singular beams through atmospheric turbulence[J]. Applied Optics, 58, 6335-6345(2019).

[31] Hill R J. Models of the scalar spectrum for turbulent advection[J]. Journal of Fluid Mechanics, 88, 541-562(1978).

[32] Andrews L C, Phillips R L[M]. Laser Beam Propagation through Random Media(2005).

[33] Luo C K, Lu F, Miao Z F et al. Propagation and spreading of radial vortex beam array in atmosphere[J]. Acta Optica Sinica, 39, 0601004(2019).

[34] He K M, Zhang X Y, Ren S Q et al. Deep residual learning for image recognition[C], 770-778(2016).

[35] Zheng Y P, Li G Y, Li Y. Survey of application of deep learning in image recognition[J]. Computer Engineering and Applications, 55, 20-36(2019).