Introduction

An optical vortex has helical phase front and doughnut-shaped intensity profile which carries orbital angular momentum (OAM) defined by the azimuthal phase dependence e il𝜃 , with l known as the topological charge (TC) of the vortex beam. [1] The OAM has an unbounded value of lℏ per photon which provides an extra degree of freedom for increasing data capacity in optical communication, [2] quantum information processing, [3,4] and security. [5] Besides this, OAM beams have found other applications including optical tweezers, [6,7] optical trapping, [8] and highresolution lithography. [9] The conventional methods of generating optical vortices by using bulky optical components such as spiral phase plates and spatial light modulators are not compatible with on-chip photonic integration. Plasmonic metasurfaces which allow arbitrary control of the phase front of an optical holograms and the generated optical vortex beams are recorded in TH far-field diffraction images. Strong anisotropic responses of the generated TH optical vortices are characterized by rotating the linear polarization of the fundamental beam. It is shown that the vortex beam with a defined topological charge can be produced with 1D fork gratings. The anisotropic nonlinear vortex beam generation is achieved independent of the fork grating orientation with respect to the GeAs crystallographic orientation. Furthermore, 2D fork hologram is used to generate multiple optical vortices having different topological charges with strong anisotropic responses. The demonstrated results offer new opportunities in advancing 2D material-based anisotropic nonlinear optical devices for future applications in photonic integrated circuits, optical communication, and optical information processing.

Results and Discussion

The top view of the in-plane lattice structure of an individual GeAs layer is illustrated in Figure 1a, which shows that GeAs has low in-plane crystal symmetry with distinguished armchair and zigzag directions. The multilayer GeAs is formed by stacking individual layers via weak van der Waals forces. This type of lowsymmetry lattice structure corresponding to two different Ge-Ge bonds along armchair and zigzag directions indicates strong inplane anisotropic optical responses in 2D GeAs crystal. For 2D GeAs crystal with monoclinic lattice structure in the space group of C2/m, the total intensity of the emitted THG signal can be represented as [47]

where 𝜒

(3) im is the third-order nonlinear susceptibility tensor, 𝜃 is the excitation polarization angle of the fundamental beam with respect to the armchair axis of the crystal, and the excitation direction is perpendicular to the lattice plane illustrated in Figure 1a. The THG emission process from GeAs crystal is first confirmed by measuring the excitation power and wavelength dependences of the THG signal generated from a GeAs flake with the thickness of 91 nm transferred on a glass substrate. The transmitted THG signal from the flake is measured using a 1560 nm pump laser and a combination of a 900 nm shortpass filter and a 520 nm bandpass filter. Figure 1b plots the measured THG output signal as a function of the input pump power in a log-log scale. The cubic power law fitting with a slope of 2.98 confirms the THG process in GeAs crystal. The wavelength dependence of the emitted THG signal is plotted in Figure 1c, showing an emission peak at 520 nm which is exactly one-third of the excitation wavelength of 1560 nm.

The fork holograms are designed to imprint the amplitude and phase information of optical vortices on the 2D GeAs crystal to generate TH optical vortex beams of TC = l in the far field. The one-dimensional fork gratings are directly milled in the 2D GeAs crystal with the binary amplitude modulation of the third-order nonlinear susceptibility given by 𝜒 (3) (x, y) = 𝜒 (3) {

where Λ is the grating period for diffraction, 𝜙(x, y) and A(x, y) are the phase and amplitude profiles of the beam wavefront in the hologram plane. The phase profile for producing TH optical vortex beam of TC = l is

and the amplitude profile A (x, y) = 1. The generated TH vortex beam appears in the first diffraction order with the diffraction angle determined by the THG wavelength and the grating period as 𝜃 THG = sin -1 (𝜆 THG ∕Λ). The binary-amplitude fork holograms are fabricated using focused ion beam (FIB) milling of the GeAs thin flakes transferred on glass substrates. The grating period is chosen to be Λ = 2.4 μm and the total FIB patterning area is 20 × 20 μm 2 . Figure 2a,b shows the optical transmission microscope image and the scanning electron microscope (SEM) image of the fabricated fork hologram to generate TH optical vortices with TC = ±1 in the first diffraction order, with the grating lines aligned with the armchair axis. The armchair axis is identified from the polarization-dependent THG measurements of the bare GeAs flake with the direction of the maximum THG intensity corresponding to the armchair axis. [47] From the atomic force microscopy (AFM) image and the line profile drawn across the flake shown in Figure 2c, the thickness of the GeAs flake is estimated to be 90 nm. The AFM line profile also reveals that the flake has been completely etched between the grating lines during the FIB process. anisotropy ratio of optical vortices is 7.19 which is similar to the value of 6.01 for the bare flake. The above measurements with the fork holograms oriented along different directions with respect to the armchair axis demonstrate the versatility of the devices considered here and predict highly anisotropic TH vortex beam generation having twofold polarization dependence pattern with the maximum THG intensity occurring along the armchair axis for an arbitrarily oriented fork hologram.

The THG conversion efficiency of the hologram 𝜂 CE is defined as the ratio of the THG power at the output and the excitation pump power. The THG conversion efficiency is measured by setting the incident linear polarization of pump beam along the armchair direction. The average pump power in the measurements is 56 mW which corresponds to a peak irradiance of 1.36 GW cm -2 . The THG power is integrated from the charge-coupled device (CCD) image of THG emission from the fork grating. The THG conversion efficiencies are determined as 3.13 × 10 -10 , 2.29 × 10 -10 , and 2.27 × 10 -10 for the 0°, 45°, and 90°oriented fork grating holograms, respectively. It is shown that the THG conversion efficiencies are almost the same for the 0°, 45°, and 90°oriented fork gratings with the flake thicknesses of 90, 80, and 75 nm, respectively. This result is consistent with the previously demonstrated optimal GeAs flake thickness range between 40 and 120 nm in terms of high THG conversion efficiency. [47] The TH vortex conversion efficiency 𝜂 vortex for the generated TH vortices at the first diffraction order is defined as the fraction of the excitation pump power converted into the first-order TH vortices at the output with 𝜂 vortex = 𝜂 CE⋅ 𝜂 DE , where 𝜂 DE represents the diffraction efficiency defined as the fraction of THG power coupled to the first diffraction order. The diffraction efficiencies for the first diffraction order are calculated as 30.5%, 24.5%, and 26.0% for the 0°, 45°, and 90°oriented fork gratings, respectively. Finally, the TH vortex conversion efficiencies for the three fork gratings are determined as 9.53 × 10 -11 , 5.60 × 10 -11 , and 5.90 × 10 -11 , respectively.

Furthermore, the demonstrated 1D fork holograms to produce anisotropic TH optical vortex beam with a defined topological charge is further extended to a 2D fork hologram to generate multiple optical vortices of different topological charges with strong anisotropic responses. Binary amplitude modulations of the third-order nonlinear susceptibility are performed along both in-plane orthogonal directions of x and y axes to obtain the 2D fork hologram, which is described as the product of two fork grating functions in Equation (2) with 𝜒

(y, x). The constructed 2D fork hologram allows independent control over the TCs of the generated TH optical vortices in each direction, with TC = l x in the x direction and TC = l y in the y direction. tio for the fork grating is changed compared with the bare flake. Furthermore, the levels of gallium ion contamination in the fork gratings are different to each other due to the varying ion milling conditions used in fabrication, which will induce different levels of influence to the THG anisotropy ratio for the fork gratings. Therefore, the THG anisotropy ratio of optical vortices is different for each fork grating. Anisotropic THG of various 2D materials belonging to different crystal systems have been studied previously with the maximum reported THG anisotropy ratios of 3.24 for BP, 1.44 for ReS 2 , 4.41 for GeSe, 5.76 for GeAs, 1.90 for SiP, and 2.79 for As 2 S 3 . [48,49] The measured THG anisotropy ratios of GeAs flakes from 3.44 to 6.19 reported here are consistent with the previously reported values for GeAs flakes with different thicknesses. [47]

Conclusion

In summary, we have designed and fabricated binary-amplitude fork holograms patterned on ultrathin 2D GeAs flakes to generate highly anisotropic TH optical vortex beams. It has been shown that the TH optical vortex beam with a defined topological charge can be produced with 1D fork gratings, while strong anisotropy in TH optical vortices can always be observed irrespective of the relative orientation of the fork grating lines with respect to the armchair direction of GeAs crystal. It has also been demonstrated that multiple TH optical vortices having different topological charges with strong anisotropic responses can be generated with a single 2D fork hologram. Strong anisotropic responses in the generated TH optical vortices are observed due to the low inplane crystal symmetry of GeAs. It is demonstrated that the THG intensities of the generated optical vortices show the two-lobe polarization dependence profiles, with the maximum and minimum intensities occurring when the incident polarization is along the armchair and zigzag axes of GeAs crystal, respectively. The demonstrated binary-amplitude nonlinear hologram design based on anisotropic 2D materials can readily be extended to generate various types of exotic nonlinear optical beams by imprinting the corresponding amplitude and phase information into the 2D crystal with the binary modulation of nonlinear susceptibility. The ability to control the topological charges of nonlinear optical vortices along with having strong anisotropic responses of the 2D material-based nonlinear optical devices considered here can find potential applications in photonic integrated circuits, optical communication, and optical information processing.

Experimental Section

Sample Preparation: GeAs thin flakes were mechanically exfoliated from a bulk GeAs crystal (2D Semiconductors) to polydimethylsiloxane (PDMS) substrates using scotch tape and subsequently transferred to glass substrates using the dry transfer technique. Before the transfer, the glass substrates were cleaned using acetone, isopropyl alcohol, and deionized water. The fork holograms were fabricated using focused ion beam milling (FEI Helios Nanolab 600) of the GeAs flakes with the patterned area of 20 × 20 μmfoot_2 . Before the focused ion beam patterning, the sample was spin coated with a thin layer of the water-soluble conductive polymer to avoid the surface charging built up on the sample. The polymer layer was removed by rinsing the sample with deionized water after patterning, and finally the sample was blown dry with nitrogen gas.

Optical Setup: Linearly polarized pump laser beam at 1560 nm (Calmer fiber laser, pulse width 90 fs, repetition rate 80 MHz) was focused on the sample using a 4× objective lens (NA = 0.12). The laser spot diameter on the sample was ≈27 μm, which illuminated the entire area of the hologram. A half-wave plate was used in the excitation path to rotate the linear polarization of the pump beam. The transmitted signal from the sample was collected with a 20× objective lens (NA = 0.42), passed through a shortpass filter (900 nm) and a bandpass filter (520 nm) to select the THG emission, and imaged either in Fourier space or real space using a color CCD camera. The THG spectrum was measured with an optical spectrometer (Horiba, iHR550).