[1] Coullet P, Gil L, Rocca F. Optical vortices[J]. Optics Communications, 73, 403-408(1989).

[2] Allen L, Beijersbergen M W. Spreeuw R J C, et al. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes[J]. Physical Review A, 45, 8185-8189(1992).

[3] Wang J, Yang J Y, Fazal I M et al. Terabit free-space data transmission employing orbital angular momentum multiplexing[J]. Nature Photonics, 6, 488-496(2012).

[4] Mair A, Vaziri A, Weihs G et al. Entanglement of the orbital angular momentum states of photons[J]. Nature, 412, 313-316(2001).

[5] Lafong A, Hossack W J, Arlt J et al. Time-multiplexed Laguerre-Gaussian holographic optical tweezers for biological applications[J]. Optics Express, 14, 3065-3072(2006).

[6] Chen Z Z, Zeng T T, Qian B J et al. Complete shaping of optical vector beams[J]. Optics Express, 23, 17701-17710(2015).

[7] Ladavac K, Grier D G. Microoptomechanical pumps assembled and driven by holographic optical vortex arrays[J]. Optics Express, 12, 1144-1149(2004).

[8] Abramochkin E, Alieva T. Closed-form expression for mutual intensity evolution of Hermite-Laguerre-Gaussian Schell-model beams[J]. Optics Letters, 42, 4032-4035(2017).

[9] Wang Y, Chen Y J, Zhang Y F et al. Generalised Hermite-Gaussian beams and mode transformations[J]. Journal of Optics, 18, 055001(2016).

[10] Bandres M A. Gutiérrez-Vega J C. Ince-Gaussian modes of the paraxial wave equation and stable resonators[J]. Journal of the Optical Society of America A, 21, 873-880(2004).

[11] Woerdemann M, Alpmann C, Denz C. Optical assembly of microparticles into highly ordered structures using Ince-Gaussian beams[J]. Applied Physics Letters, 98, 111101(2011).

[12] Tuan P H, Liang H C, Huang K F et al. Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd∶YAG/Cr 4+∶YAG laser[J]. IEEE Journal of Selected Topics in Quantum Electronics, 24, 1-9(2018).

[13] Chen Y F, Tung J C, Tuan P H et al. Symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators[J]. Annalen Der Physik, 529, 1600253(2017).

[14] Götte J B. O'Holleran K, Preece D, et al. Light beams with fractional orbital angular momentum and their vortex structure[J]. Optics Express, 16, 993-1006(2008).

[15] Leach J, Yao E, Padgett M J. Observation of the vortex structure of a non-integer vortex beam[J]. New Journal of Physics, 6, 71(2004).

[16] Nasalski W. Vortex and anti-vortex compositions of exact elegant Laguerre-Gaussian vector beams[J]. Applied Physics B, 115, 155-159(2014).

[17] Milione G, Evans S, Nolan D A et al. Higher order Pancharatnam-Berry phase and the angular momentum of light[J]. Physical Review Letters, 108, 190401(2012).

[18] Yu N, Genevet P, Kats M A et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction[J]. Science, 334, 333-337(2011).

[19] Zhang L G, Shen B F, Zhang X M et al. Deflection of a reflected intense vortex laser beam[J]. Physical Review Letters, 117, 113904(2016).

[20] Mourka A, Baumgartl J, Shanor C et al. Visualization of the birth of an optical vortex using diffraction from a triangular aperture[J]. Optics Express, 19, 5760-5771(2011).

[21] Hickmann J M. Fonseca E J S, Soares W C, et al. Unveiling a truncated optical lattice associated with a triangular aperture using light's orbital angular momentum[J]. Physical Review Letters, 105, 053904(2010).

[22] Pattanayak A K, Maji S, Brundavanam M M. Polarization singularities due to unfolding of fractional vortex beams in a birefringent crystal. [C]∥Frontiers in Optics 2017. Washington, D.C.: OSA, 91(2017).

[23] Cai X, Wang J, Strain M J et al. Integrated compact optical vortex beam emitters[J]. Science, 338, 363-366(2012).

[24] Miao P, Zhang ZF, Sun J B et al. Orbital angular momentum microlaser[J]. Science, 353, 464-467(2016).

[25] Zhang W D, Wei K Y, Huang L G et al. Optical vortex generation with wavelength tunability based on an acoustically-induced fiber grating[J]. Optics Express, 24, 19278-19285(2016).

[26] Laabs H, Ozygus B. Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping[J]. Optics & Laser Technology, 28, 213-214(1996).

[27] Chen Y F, Huang T M, Kao C F et al. Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers[J]. IEEE Journal of Quantum Electronics, 33, 1025-1031(1997).

[28] Shen Y J, Meng Y, Fu X et al. Wavelength-tunable Hermite-Gaussian modes and an orbital-angular-momentum-tunable vortex beam in a dual-off-axis pumped Yb∶CALGO laser[J]. Optics Letters, 43, 291-294(2018).

[29] Beijersbergen M W. Allen L, van der Veen H E L O, et al. Astigmatic laser mode converters and transfer of orbital angular momentum[J]. Optics Communications, 96, 123-132(1993).

[30] Chen Y F, Chang C C, Lee C Y et al. Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite-Gaussian beams[J]. Laser Physics, 28, 015002(2018).

[31] Habraken S J M, Gerard N. Orbital angular momentum in twisted and rotating cavity modes[J]. Proceedings of SPIE, 6905, 690504(2008).

[32] Habraken S J M, Nienhuis G. Rotational stabilization and destabilization of an optical cavity[J]. Physical Review A, 79, 011805(2009).

[33] Huang X X, Xu B, Cui S W et al. Direct generation of vortex laser by rotating induced off-axis pumping[J]. IEEE Journal of Selected Topics in Quantum Electronics, 24, 1-6(2018).

[34] Tuan P H, Hsieh Y H, Tu C W et al. Generating high-order transverse patterns in optically pumped semiconductor lasers[J]. IEEE Journal of Selected Topics in Quantum Electronics, 25, 1-7(2019).

[35] Chen Y F, Lu T H, Su K W et al. Devil's staircase in three-dimensional coherent waves localized on lissajous parametric surfaces[J]. Physical Review Letters, 96, 213902(2006).

[36] Lu T H, Lin Y C, Chen Y F et al. Three-dimensional coherent optical waves localized on trochoidal parametric surfaces[J]. Physical Review Letters, 101, 233901(2008).

[37] Erhard J, Laabs H, Bernd O et al. Diode-pumped multipath laser oscillators[J]. Proceedings of SPIE, 3611, 2-10(1999).

[38] Dingjan J, van Exter M P, Woerdman J P. Geometric modes in a single-frequency Nd∶YVO4 laser[J]. Optics Communications, 188, 345-351(2001).

[39] Lu T H, Lin Y C, Chen Y F et al. Generation of multi-axis Laguerre-Gaussian beams from geometric modes of a hemiconfocal cavity[J]. Applied Physics B, 103, 991-999(2011).

[40] Chen Y F, Tung J C, Chiang P Y et al. Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping[J]. Physical Review A, 88, 013827(2013).

[41] Shen Y J, Yang X L, Fu X et al. Periodic-trajectory-controlled, coherent-state-phase-switched, and wavelength-tunable SU(2) geometric modes in a frequency-degenerate resonator[J]. Applied Optics, 57, 9543-9549(2018).

[42] Chen Y F, Tuan P H, Cho C Y et al. Exploring vortex structures of circularly geometric beams from off-axis pumped solid-state lasers with an external mode converter. [C]∥Lasers Congress 2016 (ASSL, LSC, LAC), Boston, Massachusetts. Washington, D.C.: OSA, JTh2A, 14(2016).

[43] Shen Y J, Wan Z S, Meng Y et al. Generation of polygonal vortex beams in quasi-frequency- degenerate states of Yb: CALGO laser. [C]∥Laser Congress 2018 (ASSL), Boston, Massachusetts. Washington, D.C.: OSA, AW2A, 3(2018).

[44] Chen Y F, Lan Y P. Formation of optical vortex lattices in solid-state microchip lasers: spontaneous transverse mode locking[J]. Physical Review A, 64, 063807(2001).

[45] Shen Y J, Wan Z S, Fu X et al. Vortex lattices with transverse-mode-locking states switching in a large-aperture off-axis-pumped solid-state laser[J]. Journal of the Optical Society of America B, 35, 2940-2944(2018).

[46] Niziev V G, Chang R S, Nesterov A V. Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer[J]. Applied Optics, 45, 8393-8399(2006).

[47] Lin Y Y, Yeh C C. Optical vortex beam conversion based on resonator with an intra-cavity spiral phase plate. [C]∥2017 Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR), July 31-August 4, 2017. New York: IEEE, s1641(2017).

[48] Oron R, Danziger Y, Davidson N et al. Laser mode discrimination with intra-cavity spiral phase elements[J]. Optics Communications, 169, 115-121(1999).

[49] Kim D J, Kim J W. High-power TEM00 and Laguerre-Gaussian mode generation in double resonator configuration[J]. Applied Physics B, 121, 401-405(2015).

[50] Kim D J. MacKenzie J I, Kim J W. Adaptable beam profiles from a dual-cavity Nd∶YAG laser[J]. Optics Letters, 41, 1740-1743(2016).

[51] Marrucci L. The q-plate and its future[J]. Journal of Nanophotonics, 7, 078598(2013).

[52] Konforti N, Marom E, Wu S T. Phase-only modulation with twisted nematic liquid-crystal spatial light modulators[J]. Optics Letters, 13, 251-253(1988).

[53] Naidoo D, Roux F S, Dudley A et al. Controlled generation of higher-order Poincaré sphere beams from a laser[J]. Nature Photonics, 10, 327-332(2016).

[54] Litvin I A, Ngcobo S, Naidoo D et al. Doughnut laser beam as an incoherent superposition of two petal beams[J]. Optics Letters, 39, 704-707(2014).

[55] Ngcobo S, Litvin I, Burger L et al. A digital laser for on-demand laser modes[J]. Nature Communications, 4, 2289(2013).

[56] Liu S S, Chen X D, Pu J X et al. A V-folded digital laser for on-demand vortex beams by astigmatic transformation of hermite-Gaussian modes[J]. Chinese Physics Letters, 36, 124203(2019).

[57] Sato T, Kozawa Y, Sato S. Transverse-mode selective laser operation by unicursal fast-scanning pumping[J]. Optics Letters, 40, 3245-3248(2015).

[58] Litvin I A, King G, Strauss H. Beam shaping laser with controllable gain[J]. Applied Physics B, 123, 174(2017).

[59] Ito A, Kozawa Y, Sato S. Generation of hollow scalar and vector beams using a spot-defect mirror[J]. Journal of the Optical Society of America A, 27, 2072-2077(2010).

[60] Schepers F, Bexter T, Hellwig T et al. Cavity-external spatial gain shaping for selective higher-order mode excitation. [C]∥Conference on Lasers and Electro-Optics, San Jose, California. Washington, D.C.: OSA, JTu2A, 61(2019).

[61] Chen D M, Miao Y J, Fu H et al. High-order cylindrical vector beams with tunable topological charge up to 14 directly generated from a microchip laser with high beam quality and high efficiency[J]. APL Photonics, 4, 106106(2019).

[62] Sueda K, Miyaji G, Miyanaga N et al. Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses[J]. Optics Express, 12, 3548-3553(2004).

[63] Lee W M, Yuan X C, Cheong W C. Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation[J]. Optics Letters, 29, 1796-1798(2004).

[64] Kotlyar V V, Elfstrom H, Turunen J et al. Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate[J]. Journal of the Optical Society of America A, 22, 849-861(2005).

[65] Marrucci L, Manzo C, Paparo D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media[J]. Physical Review Letters, 96, 163905(2006).

[66] Karimi E, Piccirillo B, Marrucci L et al. Light propagation in a birefringent plate with topological charge[J]. Optics Letters, 34, 1225-1227(2009).

[67] Piccirillo B. D'Ambrosio V, Slussarenko S, et al. Photon spin-to-orbital angular momentum conversion via an electrically tunableq-plate[J]. Applied Physics Letters, 97, 241104(2010).

[68] Miyamoto K, Suizu K, Akiba T et al. Direct observation of the topological charge of a terahertz vortex beam generated by a Tsurupica spiral phase plate[J]. Applied Physics Letters, 104, 261104(2014).

[69] Peele A G. McMahon P J, Paterson D, et al. Observation of an X-ray vortex[J]. Optics Letters, 27, 1752-1754(2002).

[70] Moh K J, Yuan X C, Cheong W C et al. High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate[J]. Applied Optics, 45, 1153-1161(2006).

[71] Nivas J J J, He S T, Rubano A et al. Direct femtosecond laser surface structuring with optical vortex beams generated by a q-plate[J]. Scientific Reports, 5, 17929(2015).

[72] Hsueh C K, Sawchuk A A. Computer-generated double-phase holograms[J]. Applied Optics, 17, 3874-3883(1978).

[73] Arrizón V. Sánchez-De-la-llave D. Double-phase holograms implemented with phase-only spatial light modulators: performance evaluation and improvement[J]. Applied Optics, 41, 3436-3447(2002).

[74] Gibson G, Courtial J, Padgett M J et al. Free-space information transfer using light beams carrying orbital angular momentum[J]. Optics Express, 12, 5448-5456(2004).

[75] Litvin I A, Dudley A, Roux F S et al. Azimuthal decomposition with digital holograms[J]. Optics Express, 20, 10996-11004(2012).

[76] Bentley J B, Davis J A, Bandres M A et al. Generation of helical Ince-Gaussian beams with a liquid-crystal display[J]. Optics Letters, 31, 649-651(2006).

[77] Mendoza-Yero O, Mínguez-Vega G, Lancis J. Encoding complex fields by using a phase-only optical element[J]. Optics Letters, 39, 1740-1743(2014).

[78] Arrizón V, Ruiz U, Carrada R et al. Pixelated phase computer holograms for the accurate encoding of scalar complex fields[J]. Journal of the Optical Society of America A, 24, 3500-3507(2007).

[79] Forbes A, Dudley A. McLaren M. Creation and detection of optical modes with spatial light modulators[J]. Advances in Optics and Photonics, 8, 200-227(2016).

[80] Rosales-Guzmán C, Bhebhe N, Mahonisi N et al. Multiplexing 200 spatial modes with a single hologram[J]. Journal of Optics, 19, 113501(2017).

[81] López-Mariscal C, Helmerson K. Shaped nondiffracting beams[J]. Optics Letters, 35, 1215-1217(2010).

[82] Ma H X, Li X Z, Tai Y P et al. Generation of circular optical vortex array[J]. Annalen Der Physik, 529, 1700285(2017).

[83] Wan Z S, Shen Y J, Gong M L et al. Quadrant-separable multi-singularity vortices manipulation by coherent superposed mode with spatial-energy mismatch[J]. Optics Express, 26, 34940-34955(2018).

[84] O'NEIL A T. COURTIAL J. Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter[J]. Optics communications, 181, 35-45(2000).

[85] Chu S C, Yang C S, Otsuka K. Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer[J]. Optics Express, 16, 19934-19949(2008).

[86] Vyas S, Kozawa Y, Sato S. Polarization singularities in superposition of vector beams[J]. Optics Express, 21, 8972-8986(2013).

[87] Liu S, Qi S X, Zhang Y et al. Highly efficient generation of arbitrary vector beams with tunable polarization, phase, and amplitude[J]. Photonics Research, 6, 228-233(2018). http://www.opticsjournal.net/Articles/Abstract?aid=OJ180330000160lSoUrX

[88] Lu T H, Huang T D, Wang J G et al. Generation of flower high-order Poincaré sphere laser beams from a spatial light modulator[J]. Scientific Reports, 6, 39657(2016).

[89] Rosales-Guzmán C, Bhebhe N, Forbes A. Simultaneous generation of multiple vector beams on a single SLM[J]. Optics Express, 25, 25697-25706(2017).

[90] Meinzer N, Barnes W L, Hooper I R. Plasmonic meta-atoms and metasurfaces[J]. Nature Photonics, 8, 889-898(2014).

[91] Liu Z T, Meng Y, Hu F T et al. Largely tunable terahertz circular polarization splitters based on patterned graphene nanoantenna arrays[J]. IEEE Photonics Journal, 11, 1-11(2019).

[92] Meng Y, Hu F T, Liu Z T et al. Chip-integrated metasurface for versatile and multi-wavelength control of light couplings with independent phase and arbitrary polarization[J]. Optics Express, 27, 16425-16439(2019).

[93] Li G X, Kang M, Chen S M et al. Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light[J]. Nano Letters, 13, 4148-4151(2013).

[94] Karimi E. Schulz S A, de Leon I, et al. Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface[J]. Light: Science & Applications, 3, e167(2014).

[95] Ma X L, Pu M B, Li X et al. A planar chiral meta-surface for optical vortex generation and focusing[J]. Scientific Reports, 5, 10365(2015).

[96] Yang K P, Pu M B, Li X et al. Wavelength-selective orbital angular momentum generation based on a plasmonic metasurface[J]. Nanoscale, 8, 12267-12271(2016).

[97] Devlin R C, Ambrosio A, Wintz D et al. Spin-to-orbital angular momentum conversion in dielectric metasurfaces[J]. Optics Express, 25, 377-393(2017).

[98] Li Y, Li X, Chen L W et al. Orbital angular momentum multiplexing and demultiplexing by a single metasurface[J]. Advanced Optical Materials, 5, 1600502(2017).

[99] Yue F Y, Wen D D, Zhang C M et al. Multichannel polarization-controllable superpositions of orbital angular momentum states[J]. Advanced Materials, 29, 1603838(2017).

[100] Devlin R C, Ambrosio A, Rubin N A et al. Arbitrary spin-to-orbital angular momentum conversion of light[J]. Science, 358, 896-901(2017).

[101] Molesky S, Lin Z, Piggott A Y et al. Inverse design in nanophotonics[J]. Nature Photonics, 12, 659-670(2018).

[102] Xie Z W, Lei T, Li F et al. Ultra-broadband on-chip twisted light emitter for optical communications[J]. Light: Science & Applications, 7, 18001(2018).