Vector beams with space-varying polarization distributions are the solution of the vector paraxial wave equation. From a historical perspective, research on the vector optical field can be traced back to fifty years ago ^[1]. Ever since the original concept was first pioneered by Snitzer ^[1], vector optical beams have attracted tremendous attention and academic interest because of their potential applications in various research realms, such as super-resolution imaging ^[2], optical trapping ^[3,4], and laser micromachining ^[5,6]. Prominent paradigms of vector optical beams are radially and azimuthally polarized beams. Due to the sharply focused light below the diffraction limit, radially polarized beams have attracted a great deal of interest in laser machining ^[7,8] and particle acceleration ^[9,10]. On the contrary, the azimuthally polarized beam can induce a strong longitudinal magnetic field, which provides potential application in probing magnetic interactions ^[2,11]. Besides their applications in classical optics, unusual attributes of the vector beams are especially interesting in quantum mechanics foundations such as generating novel cluster states ^[12] and quantum memory ^[13]. Motivated by those applications, a variety of methods have been used to generate the vector optical beams. One direct way is by utilizing a Yb fiber laser incorporating an intracavity axicon ^[14] or a conical Brewster prism ^[15]. The indirect methods employed to generate the vector optical beams are shaping the wavefront with the aid of spatial light modulators ^[16–19], metasurfaces ^[20], q-plates ^[21,22], digital micromirror devices ^[23], and so on. However, the intensity profile or the radius of the traditional vector beams is strongly dependent on the absolute value of topological charge (TC), which makes it difficult to couple into a fiber. Besides, for the vector beam optical manipulation fields, the radius changing of the traditional vector beams with different TC is also not a stable way to control the particles’ trajectory. Recently, considerable attention was paid to the generation of the perfect vector (PV) and elliptic PV beams, where the radius of the intensity profile is independent of the TC ^[22,24–28]. Based on the digitalized geometric phases ^[22], the dynamic modulation of the geometric phases ^[24,25], and the phase elements and interferometer ^[26,27], PV beams are generated successfully in the experiment. However, all these methods are actually conducted in linear optics. The generation and manipulation of such PV beams in nonlinear optical fields are still lacking. Recently, nonlinear generation and manipulation of vector beams in frequency conversion processes have been successfully implemented both in experimental and simulated environments ^[19,29].

In this study, we propose a method to generate the PV beams both in a fundamental frequency (FF) and a second harmonic (SH) wavebands at the same time. In our experiment, PV beams with different polarization states are generated. In addition, we measure the vectorial properties of the generated PV beams by adding a Glan–Taylor (GT) prism before the charge coupled device (CCD). The experiment results are consistent with the theoretical simulations. Moreover, the results demonstrated that the radii of the generated PV beams are equal both in linear and nonlinear wavebands which are independent of the TC.

The experimental setup is illustrated in Fig. 1. The high-energy diode-pumped all-solid-state Q-switched laser system delivers linearly polarized 10 ns pulses at a central wavelength of 1064 nm and operates at a 1 kHz repetition. A GT prism is used to adjust the FF beam to the horizontal polarization. A half wave plate (HWP) and a quarter wave plate (QWP1) with its fast axis arranged along 45° with respect to the horizontal direction are employed to adjust the polarization state of the FF wave. The FF wave is split into two orthogonal directions by a polarized beam splitter (PBS). Three mirrors (M1, M2, M3) and a spiral phase plate (SPP) are composed of a Sagnac interferometer. The SPP is located at the place where two separated beams have same optical length before incident into it. Two orthogonal beams are recombined by a PBS after which the FF wave can be expressed in the form of where Eω represents the FF wave. A and l refer to the amplitude and TC of the FF wave, respectively. φ describes the azimuthal angle, and δ is the phase difference. γ denotes the angle between the fast axis of HWP and the horizontal direction. After passing through an axicon whose base angle is 2°, the Bessel beam in the cylindrical coordinate system (ρ,φ,z) is generated, which can be written as where Jl refers to the first kind of Bessel function. kr and kz, respectively, refer to the wave vectors along radial and propagation directions and satisfy the relationship of k=kr2+kz2=2π/λ, where λ is the wavelength of the incident FF wave. Theoretically, the perfect vortex beam is the Fourier transformation of a Bessel Gaussian beam ^[30]. Therefore, a lens (L1) placed before QWP2 with focus length (f) of 200 mm is employed to perform the Fourier transformation in our experiment. According to the property of the Fourier transformation and the orthogonality of the Bessel functions, the form of the FF wave in the polar coordinate system (r,θ) can be derived as Another QWP2 with a fast axis along −45° is employed to convert the two linearly polarized beams into PV beams, which can be simplified as Using the method proved above, the PV beam at the FF waveband is generated. In order to generate nonlinear PV beams, two cascading nonlinear crystals are inserted between the axicon and the dichroic mirror (DM). Both samples (S) are 5% (mole fraction) MgO:LiNbO3 crystals, and they are placed in two orthogonal directions. Under the undepleted pump and paraxial approximation, we have an equation that describes the SH optical field as where E2ω represents the generated SH beam. ω, deff, c, and n2ω are, respectively, the angular frequency of the FF beam, the effective nonlinear coefficient, the light speed in vacuum, and the refractive index of the generated SH beam. In our experiment, the Type-I (oo-e) phase matching condition is satisfied. Finally, the generated SH optical field can be written as L0 is the length of the nonlinear crystal along the propagation direction. After substituting Eq. (2) into Eq. (6), we can get After passing through a Fourier transformation lens L2 and a QWP3, the generated SH PV beam in the polar coordinate system (r,θ) can be written as Here, the fast axis of the QWP3 is arranged along a 45° direction with respect to the horizontal direction. At last, two spatial filters (SF1 and SF2) are used to pick out the PV beams. CCDs located at the Fourier plane of the lenses L1 and L2 are used to record the generated PV beam. The results of generated FF and SH PV beams are, respectively, shown in Figs. 1(a) and 1(b). To make the figures vivid, we use red as the pseudo color to show the intensity distribution of the FF beam.

First, the radially polarized PV beam is studied. Figure 2 displays the simulated and experimental results of the FF PV beams with TC l=1. In this situation, the initial phase δ/2+2γ=0 and the generated FF PV beams can be described as Eω(r,θ)=2Aδ(r−krf/k)exp(iδ/2)(cos(lθ)sin(lθ)). Figure 2(a) theoretically describes the polarization and intensity distributions of the FF PV beam. The corresponding experimental result is presented in Fig. 2(b). To characterize the vectorial property of the PV beam, a GT prism is inserted into the optical path between SF1 and CCD1. As shown in Figs. 2(a1)–2(a9) and Figs. 2(b1)–2(b9), the intensity pattern of the PV beams changes with the polarization angle of the GT prism ranging from 0° to 160°. It is clear that when the GT prism has a polarization angle of 0°, the signal of the PV beam presents a maximum in the horizontal direction, while a minimum at the vertical position, as shown in Figs. 2(a1) and 2(b1). When the polarization angle rotates from 0° to 80°, two crescent moon patterns gradually rotate to vertical direction, as shown in Figs. 2(a1)–2(a5) and Figs. 2(b1)–2(b5). Obviously, two crescent moon patterns appear almost in the vertical direction when the polarization angle arrives at 80° and 100° for most perpendicular components parallel to the vertical directions, as shown in Figs. 2(a5), 2(a6), and Figs. 2(b5) and 2(b6). Also, as illustrated in Figs. 2(a7)–2(a9) and Figs. 2(b7)–2(b9), two crescent moon patterns rotate to the horizontal direction as well when the polarization angle of the GT prism increases. In this whole process, the extinction direction rotates along the same direction and is always perpendicular to the polarization direction of the GT prism.

To generate the nonlinear PV beams, two orthogonal cascading nonlinear crystals with a length of 3 mm along the propagation direction of light are positioned at the Fourier plane of the axicon and lens L2, as shown in Fig. 1. Figure 3(a) shows the polarization and intensity distribution of the corresponding simulated result when 4γ+δ−π/4=0. In this case, the generated SH PV beam can be written as E2ω(r,θ)=2iωdeffcn2ωL0A2δ(r−krf/k)exp[i(δ+π/2)](cos(2lθ)sin(2lθ)). The TC of the generated SH PV beams is twice that of the FF ones owing to the Type-I (oo-e) phase matching process. The generated experimental result is shown in Fig. 3(b). The intensity pattern changes as the GT prism rotates as well. Both theoretical and experimental results are, respectively, exhibited in Figs. 3(a1)–3(a9) and Figs. 3(b1)–3(b9) as the angle of the GT prism is equal to 0°, 20°, 40°, 60°, 80°, 100°, 120°, 140°, and 160°. It can be observed that the pattern of the generated SH PV beam is split into four crescent moon patterns, and it rotates θ while the GT prism rotates 2θ. Moreover, we measured the intensity of the FF and SH beams corresponding to 13.5 mW and 0.05 mW, respectively. So the conversion efficiency is approximately 0.37%.

It is interesting to note that by simply rotating the axis direction of the HWP (γ), we can control the flexibility of the polarization states of the PV beams in a dual waveband. For simplicity and without loss of generality, two examples with δ/2+2γ=π/3 and δ/2+2γ=2π/3 are demonstrated. In this situation, the generated FF PV beams can be, respectively, simplified as Eω(r,θ)∝δ(r−krf/k)[cos(lθ+π/3),sin(lθ+π/3)] and Eω(r,θ)∝δ(r−krf/k)[cos(lθ+2π/3),sin(lθ+2π/3)]. The corresponding SH PV beams can be written as E2ω(r,θ)∝δ(r−krf/k)[cos(2lθ+5π/12),sin(2lθ+5π/12)] and E2ω(r,θ)∝δ(r−krf/k)[cos(2lθ+13π/12),sin(2lθ+13π/12)]. Figure 4 shows the corresponding experimental results. The first and third rows depict the FF PV beams, and the corresponding SH PV beams are displayed in the second and fourth rows. Note that the generated SH PV beams are completely different from the FF ones as Type-I (oo-e) nonlinear processes occur. It is clear that the two crescent moon patterns of the FF PV beams are located at different positions while the axis direction of the HWP varies as shown in the first and third rows in Fig. 4. The direction of the crescent moon patterns rotates along the same orientation of the GT prism. At the same time, the vectorial properties of the generated SH PV beams are also changed. Four crescent moon patterns appear and rotate while the angle of the GT prism changes.

At last, we measure the radius of the generated PV beams at the focus of the lenses L1 and L2, and the results are displayed in Fig. 5. It can be observed that the radii of the FF PV beams and SH PV beams are almost equal whenever the TC changes. In fact, the radius of the generated PV beam can be expressed as rr=fsin[(n−1)α], where f is the focus length of L1 or L2, and n and α refer to the refractive index and base angle of the axicon, respectively. Obviously, the radius of the generated PV beams can be undoubtedly affected by the parameters of the lens and axicon. The experimental results show that the radius of the PV beams almost keep consistent with the simulated ones and are independent of the TC. In addition, we can choose proper parameters of the lens or axicon to fulfill the demand for the different radius.

In general, the method proposed above provides an alternative route to access PV beams at different wavelengths at the same time. Our studies unambiguously confirm the generation of FF and SH PV beams, which may further the generation of PV beams in ultraviolet regimes. Furthermore, this technique has potential use in other nonlinear processes such as sum-frequency generation and difference-frequency generation.

In summary, we propose a dual waveband generator of the PV beams and demonstrate the generation of the PV beams in FF and SH wavebands at the same time. By using this setup, the polarization states of the generated PV beams can be adjusted flexibly by rotating the axis direction of the HWP. The experimental results agree well with the simulated ones. In addition, the radii of the PV beams with different TC are measured, and the results validate that they are independent of the TC. This work provides a simple way to flexibly generate PV beams with different polarization states in a dual waveband, which may have further applications in optical manipulation, light-sheet microscopy, and so on.