I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. THEORYIII. EXPERIMENTAL SETUPIV. RESULTS AND DISCUSSIO...V. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESVector beams are characterized by a spatially inhomogeneous state of polarization, e.g., the radial and azimuthal polarization states. Optical vector beams have received increasing attention, thanks to their interesting properties that are of great potential for applications such as optical tweezers,11. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt. Lett. 22, 52–54 (1997). https://doi.org/10.1364/ol.22.000052 information transmission and processing,2,32. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009). https://doi.org/10.1103/PhysRevLett.103.0136013. E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010). https://doi.org/10.1103/physreva.82.022115 and laser surface structuring.4,54. J. JJ Nivas, S. He, A. Rubano, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Direct femtosecond laser surface structuring with optical vortex beams generated by a q-plate,” Sci. Rep. 5, 17929 (2015). https://doi.org/10.1038/srep179295. E. Allahyari, J. JJ Nivas, F. Cardano, R. Bruzzese, R. Fittipaldi, L. Marrucci, D. Paparo, A. Rubano, A. Vecchione, and S. Amoruso, “Simple method for the characterization of intense Laguerre-Gauss vector vortex beams,” Appl. Phys. Lett. 112, 211103 (2018). https://doi.org/10.1063/1.5027661 Vector beams bearing high-order topological charges66. D. Chen, Y. Miao, H. Fu, H. He, J. Tong, and J. Dong, “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,” APL Photonics 4, 106106 (2019). https://doi.org/10.1063/1.5119789 are promising for quantum entanglement77. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012). https://doi.org/10.1126/science.1227193 and high-resolution imaging.88. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5, 86–92 (2018). https://doi.org/10.1364/optica.5.000086 Terahertz (THz) vector beams have found their applications in various fields, such as particle acceleration,9,109. E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015). https://doi.org/10.1038/ncomms948610. D. Zhang, M. Fakhari, H. Cankaya, A.-L. Calendron, N. H. Matlis, and F. X. Kärtner, “Cascaded multicycle terahertz-driven ultrafast electron acceleration and manipulation,” Phys. Rev. X 10, 011067 (2020). https://doi.org/10.1103/physrevx.10.011067 THz vortex dichroism spectroscopy,1111. A. A. Sirenko, P. Marsik, C. Bernhard, T. N. Stanislavchuk, V. Kiryukhin, and S.-W. Cheong, “Terahertz vortex beam as a spectroscopic probe of magnetic excitations,” Phys. Rev. Lett. 122, 237401 (2019). https://doi.org/10.1103/physrevlett.122.237401 and THz waveguide.12,1312. J. A. Deibel, K. Wang, M. D. Escarra, and D. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14, 279–290 (2006). https://doi.org/10.1364/opex.14.000279 13. Z. Zheng, N. Kanda, K. Konishi, and M. Kuwata-Gonokami, “Efficient coupling of propagating broadband terahertz radial beams to metal wires,” Opt. Express 21, 10642–10650 (2013). https://doi.org/10.1364/oe.21.010642Methods to generate infrared (IR) and visible vector beams are well established. For example, q-plates14–1714. L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011). https://doi.org/10.1088/2040-8978/13/6/06400115. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (2011). https://doi.org/10.1364/oe.19.00408516. S. Nersisyan, N. Tabiryan, D. M. Steeves, and B. R. Kimball, “Fabrication of liquid crystal polymer axial waveplates for UV-IR wavelengths,” Opt. Express 17, 11926–11934 (2009). https://doi.org/10.1364/oe.17.01192617. Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order poincaré sphere,” Photonics Res. 5, 15–21 (2017). https://doi.org/10.1364/prj.5.000015 are capable of converting linearly polarized light into vector beams bearing a topological charge of various orders. A q-plate capable of generating a vector light array was also proposed.1818. P. Chen, W. Ji, B.-Y. Wei, W. Hu, V. Chigrinov, and Y.-Q. Lu, “Generation of arbitrary vector beams with liquid crystal polarization converters and vector-photoaligned q-plates,” Appl. Phys. Lett. 107, 241102 (2015). https://doi.org/10.1063/1.4937592 Comparatively, the generation of vector beams in the THz frequency range is still in its infancy. The reported methods to generate THz vector beams include the THz antenna,1919. S. Winnerl, B. Zimmermann, F. Peter, H. Schneider, and M. Helm, “Terahertz B-Gauss beams of radial and azimuthal polarization from microstructured photoconductive antennas,” Opt. Express 17, 1571–1576 (2009). https://doi.org/10.1364/oe.17.001571 THz radial polarizer,2020. Z. Xie, J. He, X. Wang, S. Feng, and Y. Zhang, “Generation of terahertz vector beams with a concentric ring metal grating and photo-generated carriers,” Opt. Lett. 40, 359–362 (2015). https://doi.org/10.1364/ol.40.000359 and axicon Fresnel retarder.1111. A. A. Sirenko, P. Marsik, C. Bernhard, T. N. Stanislavchuk, V. Kiryukhin, and S.-W. Cheong, “Terahertz vortex beam as a spectroscopic probe of magnetic excitations,” Phys. Rev. Lett. 122, 237401 (2019). https://doi.org/10.1103/physrevlett.122.237401 Recently, we have found that a one-color filament radiates radially polarized THz pulses besides elliptically polarized ones.2121. S. Mou, A. D’Arco, L. Tomarchio, M. D. Fabrizio, A. Curcio, S. Lupi, and M. Petrarca, “Simultaneous elliptically and radially polarized THz from one-color laser-induced plasma filament,” New J. Phys. 23, 063048 (2021). https://doi.org/10.1088/1367-2630/ac03cd Unfortunately, all the mentioned methods are limited to generating THz vector beams bearing low-order topological charges.Segmented half-wave plates were adopted to generate THz quasi-vector beams.9,109. E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015). https://doi.org/10.1038/ncomms948610. D. Zhang, M. Fakhari, H. Cankaya, A.-L. Calendron, N. H. Matlis, and F. X. Kärtner, “Cascaded multicycle terahertz-driven ultrafast electron acceleration and manipulation,” Phys. Rev. X 10, 011067 (2020). https://doi.org/10.1103/physrevx.10.011067 In addition, a segmented nonlinear crystal pumped with a linearly polarized pump was also proposed to generate quasi-cylindrical THz vector beams.22,2322. R. Imai, N. Kanda, T. Higuchi, Z. Zheng, K. Konishi, and M. Kuwata-Gonokami, “Terahertz vector beam generation using segmented nonlinear optical crystals with threefold rotational symmetry,” Opt. Express 20, 21896–21904 (2012). https://doi.org/10.1364/oe.20.02189623. M. J. Cliffe, D. M. Graham, and S. P. Jamison, “Longitudinally polarized single-cycle terahertz pulses generated with high electric field strengths,” Appl. Phys. Lett. 108, 221102 (2016). https://doi.org/10.1063/1.4953024 The shortage of these segment methods is that each segment’s THz polarization is constant and homogeneous. In contrast, a real vector beam requires continuous polarization variation vs the azimuthal angle. A new emerging approach is developed to generate THz vector and vortex beams by pumping a nonlinear crystal with vector pump beams. The THz from 110-cut ZnTe pumped with a radially polarized IR beam2424. A. A. Dhaybi, J. Degert, E. Brasselet, E. Abraham, and E. Freysz, “Terahertz vortex beam generation by infrared vector beam rectification,” J. Opt. Soc. Am. B 36, 12–18 (2019). https://doi.org/10.1364/josab.36.000012 was converted into a vortex beam with a quarter-wave plate and a polarizer. A quasi-cylindrical THz vector beam was generated by pumping a nonlinear crystal with a quasi-vector pump.1313. Z. Zheng, N. Kanda, K. Konishi, and M. Kuwata-Gonokami, “Efficient coupling of propagating broadband terahertz radial beams to metal wires,” Opt. Express 21, 10642–10650 (2013). https://doi.org/10.1364/oe.21.010642

The new emerging approach exhibits the possibility of generating real THz vector beams bearing tailored topological charges by pumping a nonlinear crystal with real vector pump beams. In this article, we theoretically illustrate the relation between the pump’s topological charge and the THz’s topological charge, providing guidance on how to design the topological charge of a THz vector beam. We find that the topological charge of the THz vector beam is twice that of the IR vector pump and exhibits an opposite sign. In addition, the polarization state of the THz vector beam is controllable by manipulating the initial polarization angle. We also experimentally demonstrate the generation of THz vector beams by pumping a 111-cut ZnTe crystal with an IR vector beam.

II. THEORY

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. THEORY <<III. EXPERIMENTAL SETUPIV. RESULTS AND DISCUSSIO...V. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESThe orientation angle α of the fast axis of a liquid crystal molecule in the liquid-crystal-polymer half-wave vortex retarder (VR) bearing a topological charge lVR can be written as2525. L. Marrucci, “Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Mol. Cryst. Liq. Cryst. 488, 148–162 (2008). https://doi.org/10.1080/15421400802240524where α0,VR is the orientation of the fast axis at ϕ = 0 and ϕ denotes the azimuthal angle relative to the positive x direction in the zx plane of the laboratory coordinate system as shown in Fig. 1. The Jones matrix MVR describing the action of the liquid-crystal-polymer half-wave VR on IR beam reads2525. L. Marrucci, “Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Mol. Cryst. Liq. Cryst. 488, 148–162 (2008). https://doi.org/10.1080/15421400802240524MVR=cos2α(ϕ)sin2α(ϕ)sin2α(ϕ)−cos2α(ϕ).(2)Horizontally polarized IR beam can be expressed as E0,IR[1,0]T, with E0,IR standing for the amplitude of the IR electric field. The application of the VR on the horizontally polarized IR can be described as MVR×E0,IR[1,0]T=E0,IR[cos(2lVRϕ+2α0,VR),sin(2lVRϕ+2α0,VR)]T. Assuming ϕ0,IR = 2α0,VR and lIR = 2lVR, it can also be written asE⃗IR=E0,IRcos(lIRϕ+ϕ0,IR)sin(lIRϕ+ϕ0,IR).(3)Equation (3) represents an IR beam whose polarization angle ϕIR = lIRϕ + ϕ0,VR is a function of the azimuthal angle ϕ, indicating that the output IR beam is a vector beam, with the initial polarization angle ϕ0,IR = 2α0,VR at ϕ = 0 and topological charge lIR = 2lVR. For a vector beam, the topological charge determines the cycle number of polarization rotation with the azimuthal angle.In terms of generating linearly polarized THz for general applications, such as THz spectroscopy, a 110-cut ZnTe crystal is commonly employed due to its higher THz generation efficiency than a 111-cut ZnTe crystal.2626. Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001). https://doi.org/10.1364/josab.18.000823 However, a 111-cut ZnTe crystal is favorable for generating THz vector and vortex beams since a 111-cut ZnTe crystal can generate a pure THz vector beam when pumped with a vector beam, as illustrated in the following. In contrast, the 110-cut generates a mixture of components exhibiting different polarization states and demands a very sophisticated method to obtain a pure THz vector and vortex beam.2424. A. A. Dhaybi, J. Degert, E. Brasselet, E. Abraham, and E. Freysz, “Terahertz vortex beam generation by infrared vector beam rectification,” J. Opt. Soc. Am. B 36, 12–18 (2019). https://doi.org/10.1364/josab.36.000012 The orientation of the 111-cut ZnTe crystal is shown in Fig. 1. The propagation direction of the IR pump and the generated THz beam is in the y direction, which is normal to the crystal surface. The [−1, −1, 2] direction of the crystal is in the x direction. In Fig. 1, the direction vector near each axis is expressed with the crystallographic coordinates. A detailed description of the THz generation process is presented in the supplementary material. Here, we briefly describe the main properties of THz generation from a 111-cut ZnTe crystal. The THz generation exhibits the following characteristics: with a linearly polarized pump whose polarization angle is ϕIR, the polarization angle of the generated THz isEquation (4) shows that the polarization directions of the IR pump and the generated THz are on the opposite sides of the [−1, −1, 2] direction of the crystal. The absolute value of the angle between the THz polarization and the [−1, −1, 2] direction of the crystal is twice the value between the IR polarization and the [−1, −1, 2] direction,2626. Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001). https://doi.org/10.1364/josab.18.000823 which is also illustrated in Fig. 1.Thus, when pumped with a IR vector beam described by Eq. (3), the generated THz can be expressed as E⃗THz=E0,THz[cos(−2lIRϕ−2ϕ0,IR),sin(−2lIRϕ−2ϕ0,IR)]T, with E0,THz denoting the amplitude of THz electric field. Defining lTHz = −2lIR = −4lVR and ϕ0,THz = −2ϕ0,IR = −4ϕ0,VR, it can be written asE⃗THz=E0,THzcos(lTHzϕ+ϕ0,THz)sin(lTHzϕ+ϕ0,THz).(5)Similar to Eq. (3), Eq. (5) describes a THz vector beam whose polarization angle ϕTHz = lTHzϕ + ϕ0,THz varies with the azimuthal angle ϕ, with the initial polarization angle ϕ0,THz = −4α0,VR at ϕ = 0 and topological charge lTHz = −4lVR. The dependence of the initial polarization angles of the IR pump and THz on the initial orientation angle of the VR and the relations of topological charges of the VR, IR, and THz are summarized in Table I.

TABLE I. Dependence of the IR pump and THz’s initial polarization angles on the VR orientation and the relations of topological charges.

Initial polarizationTopological chargeϕ0,IR = 2α0,VR = θlIR = 2lVRϕ0,THz = −2ϕ0,IR = −4α0,VR = −2θlTHz = −2lIR = −4lVRIt is worth noticing that the projection of the orbital angular momentum (OAM) operator along the direction of propagation of the beam is defined as L̂y=−iℏ∂ϕ.2727. A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011). https://doi.org/10.1364/oe.19.009714 The eigenstates of such operators are of the kind eilϕ/2π, where the eigenvalue l coincides with the topological charge. The phase associated with the eigenstates is a geometric phase.28,2928. Z. X. Shen, M. J. Tang, P. Chen, S. H. Zhou, S. J. Ge, W. Duan, T. Wei, X. Liang, W. Hu, and Y. Q. Lu, “Planar terahertz photonics mediated by liquid crystal polymers,” Adv. Opt. Mater. 8, 1902124 (2020). https://doi.org/10.1002/adom.20190212429. G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order P-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012). https://doi.org/10.1103/physrevlett.108.190401 The THz electric field of Eq. (5) can be expressed through the L̂y eigenstates as follows:E⃗THz=E0,THzexp[i(lTHzϕ+ϕ0,THz)]+exp[−i(lTHzϕ+ϕ0,THz)]2exp[i(lTHzϕ+ϕ0,THz)]−exp[−i(lTHzϕ+ϕ0,THz)]2i.(6)Considering Eq. (6), the average value of the topological charge for the THz electric field is defined asl̄THz=∫02πE⃗THz*⋅L̂yE⃗THzdϕ∫02πE⃗THz*⋅E⃗THzdϕ=0.(7)Therefore, the average topological charge associated with the THz field expressed by Eq. (5) is zero. Indeed, the field consists of OAM operator eigenstates carrying opposite eigenvalues lTHz and −lTHz. This result is consistent with the literature.24,3024. A. A. Dhaybi, J. Degert, E. Brasselet, E. Abraham, and E. Freysz, “Terahertz vortex beam generation by infrared vector beam rectification,” J. Opt. Soc. Am. B 36, 12–18 (2019). https://doi.org/10.1364/josab.36.00001230. A. Curcio, S. Mou, L. Palumbo, S. Lupi, and M. Petrarca, “Selection rules for the orbital angular momentum of optically produced THz radiation,” Opt. Lett. 46, 1514–1517 (2021). https://doi.org/10.1364/ol.416814 However, since each vector beam simultaneously carries two eigenstates of L̂y exhibiting topological charges of opposite signs and the same absolute value, suitable techniques can be implemented to obtain a single eigenstate of L̂y.3131. R. Imai, N. Kanda, T. Higuchi, K. Konishi, and M. Kuwata-Gonokami, “Generation of broadband terahertz vortex beams,” Opt. Lett. 39, 3714–3717 (2014). https://doi.org/10.1364/ol.39.003714

III. EXPERIMENTAL SETUP

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. THEORYIII. EXPERIMENTAL SETUP <<IV. RESULTS AND DISCUSSIO...V. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESThe experimental setup is sketched in Fig. 2. The laser pulse from a chirped-pulse-amplification system (Legend) with a pulse duration of 50 fs, repetition frequency of 1 kHz, pulse energy of 2 mJ, and central wavelength of 800 nm is split into two parts by a beam splitter. The transmitted part through the splitter is employed as a pump to generate THz from a 3 mm-thick 111-cut ZnTe crystal. Before pumping the crystal, the linearly polarized infrared (IR) beam is converted into a vector beam with a liquid-crystal-polymer half-wave VR (Thorlabs:WPV10L-780). When pumped by the IR vector beam, the 111-cut ZnTe crystal generates a THz vector beam. The THz from the crystal transmits through a piece of Teflon, which blocks the residual IR pump. A piece of a silicon wafer further filters out residual IR. Then, a polarizer decomposes the THz vector beam into two orthogonal components. In the experiment, the two orthogonal components are, respectively, polarized in the x and z directions of the lab coordinate system.The intensity modes of the two orthogonal THz components are recorded by a THz imaging system.3232. M. Di Fabrizio, A. D’Arco, S. Mou, L. Palumbo, M. Petrarca, and S. Lupi, “Performance evaluation of a THz pulsed imaging system: Point spread function, broadband THz beam visualization and image reconstruction,” Appl. Sci. 11, 562 (2021). https://doi.org/10.3390/app11020562 An aperture with a diameter of around 1 mm is mounted on a two-dimensional stage to measure the THz intensity vs spatial position. The stage can move the aperture in the xz plane. The electric field of the THz passing through the aperture is measured with electro-optic sampling (EOS), which is described in detail elsewhere.2121. S. Mou, A. D’Arco, L. Tomarchio, M. D. Fabrizio, A. Curcio, S. Lupi, and M. Petrarca, “Simultaneous elliptically and radially polarized THz from one-color laser-induced plasma filament,” New J. Phys. 23, 063048 (2021). https://doi.org/10.1088/1367-2630/ac03cd The square of the THz electric field is integrated to represent the THz intensity. The stage moves with a step of 1 mm in both the x and z directions. The intensity is measured at each position of the aperture. Finally, THz intensity mode, i.e., THz intensity vs position, is obtained. The intensity modes of the IR pump are obtained with a CMOS camera (Thorlabs:DCC1545M).

IV. RESULTS AND DISCUSSION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. THEORYIII. EXPERIMENTAL SETUPIV. RESULTS AND DISCUSSIO... <<V. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESLiquid crystal molecules at each position of the half-wave VR work as a tiny half-wave plate. The fast axis of each tiny half-wave plate varies with the azimuthal angle as described by Eq. (1). As a half-wave plate can rotate the polarization of the input light, the VR rotates the polarization angle of the input light by a value determined by the distribution of the orientations of the fast axes of the liquid crystal molecules in the VR. In this proof-of-principle paper, we use lVR = 1/2. According to Table I, lIR = 1 and lTHz = −2 for lVR = 1/2. Besides the topological charge, we also investigate how to manipulate the initial THz polarization angle ϕ0,THz, which can be achieved by rotating the VR orientation angle θ. Figures 3(a) and 3(d), respectively, show the distribution of the fast axes of the liquid crystal molecules in the VR for the orientation angle θ = 0 and −π/4. The arrows in Figs. 3(a) and 3(d) represent the fast axes of the liquid crystals, and the two red arrows highlight the orientation of the VR. For an arbitrary θ, the distribution of the fast axes of the liquid crystal molecules in the VR can be expressed as α(ϕ) = lVRϕ + θ/2. Comparing the formula α(ϕ) = lVRϕ + θ/2 with Eq. (1) results in the relation α0,VR = θ/2. Considering the relations ϕ0,IR = 2α0,VR and ϕ0,THz = −2ϕ0,IR presented in Table I, the relation between the initial THz prioritization angle and the VR orientation angle can be expressed as ϕ0,THz = −2θ. Consequently, the initial THz prioritization angle ϕ0,THz can be manipulated by varying the VR orientation angle θ. Similarly, the initial IR prioritization angle ϕ0,IR obeys the formula ϕ0,IR = θ. In the experiment, ϕ0,IR and ϕ0,THz are varied by changing θ.For θ = 0 and lVR = 1/2, the polarization state of the IR pump can be expressed as cos(ϕ),sin(ϕ)T with ϕ0,IR = θ = 0 according to Eq. (3) and the polarization state of the generated THz can be expressed as cos(−2ϕ),sin(−2ϕ)T with ϕ0,THz = −2θ = 0 according to Eq. (5). The polarization states of the IR pump and THz for θ = 0 and lVR = 1/2 are, respectively, plotted in Figs. 3(b) and 3(c). The arrows in Figs. 3(b) and 3(c) stand for the electric field polarization, whose direction is dependent on the azimuthal angle ϕ. Similarly, for θ = −π/4 and lVR = 1/2, the polarization state of the IR pump can be expressed as cos(ϕ−π/4),sin(ϕ−π/4)T with ϕ0,IR = θ = −π/4 and the polarization state of the generated THz can be expressed as cos(−2ϕ+π/2),sin(−2ϕ+π/2)T with ϕ0,THz = −2θ = π/2. The polarization states of the IR pump and THz for θ = −π/4 and lVR = 1/2 are, respectively, plotted in Figs. 3(e) and 3(f).The previous discussions in the theory section considered the vector beams’ polarization states neglecting the intensity’s spatial dependence. Now, we proceed also to consider the spatial dependence of the intensity. The intensity profile of a vector beam can be described by the Laguerre–Gauss mode.3333. Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photonics 1, 1–57 (2009). https://doi.org/10.1364/aop.1.000001 Simultaneously considering the intensity profile and polarization, the following formula can describe a vector beam in cylindrical coordinates:E⃗(r,ϕ,y)=E0ω0ω(y)2rω(y)|l|⁡exp−r2ω2(y)Lp|l|2r2ω(y)2×exp−ikr22R(y)exp[iϕp|l|(y)]cos(lϕ+ϕ0)sin(lϕ+ϕ0),(8)where l is the topological charge of the vector beam, which can be zero or a positive or negative integer. Index p is an integer, and p ≥ 0. Lp|l|(x) denotes the associated Laguerre polynomials. ϕp|l|(y) = (2p + |l| + 1) tan−1(y/yR) is the Gouy phase. yR=πω02/λ stands for the Rayleigh range with ω0 and λ, respectively, denoting the beam waist size and wavelength. ω(y) is the beam size. k is the wavenumber, and R(y)=(y2+yR2)/y.Equation (8) provides us with a method to expe