2.1

Generation principle of the ultrafast plasmonic vortex and array

The schematic of ultrafast plasmonic vortex generation is shown in Fig. 1(a). A slit (slit width w = 50 nm) was etched on a gold film (thickness of 250 nm), and the planes of the slit on the gold film are shown as a circle and a hexagon, respectively, as shown in Fig. 1(b), which serve as plasmonic lenses for excitation and focusing of the SPP pulse [17,18]. In this work, in order to realize the ultrafast modulation of the plasmonic vortex, the incident pulsed light is regarded as a combination of two Gaussian femtosecond sub-pulses with right- and left-handed circular polarization (spin angular momenten of σ1 = +1 and σ2 = -1), respectively. The two sub-pulses have the same pulse width (wt = 100 fs) and center wavelength (λ0 = 800 nm) and partially overlap with a fixed time delay Δt = 40 fs. As a result, the polarization state of the combined pulse varies with time over the entire range of the wave packet, leading to different behaviors during the time-range of the combined pulse under the TC modulation for the plasmonic vortex field.

Figure. 1.

(a) The schematic diagram of the generation of time-varying SPP vortex field. (b) The schematic diagram of the ring-shaped slit and hexagonal slit for generation of single plasmonic vortex and arrays.

In order to compute the time-varying SPP field, we have developed an analytical model based on the Fourier transform method. For a Gaussian pulse, its frequency-domain electric field can be expressed as:

where ω0 is the center angular frequency of the pulse and wt is the pulse width. According to Fig. 1(a) and Eq. (1), the frequency-domain electric field of incident pulsed light consisting of two sub-pulses can be expressed as:

Since the longitudinal electric-field component dominates the SPP field [14,15], we further derive the z-direction electric-field components of SPPs excited by the annular slit and the hexagonal slit. The SPPs excited at each point of the annular slit propagate radially toward the center, forming an infinite number of surface wave interference SPP fields, and the optical field satisfies the Bessel function distribution in the radial direction. And each side of the hexagonal slit can excite SPP, forming six surface wave interference SPP fields, and the light field is a sixfold symmetric array light field.

where (ρ,ϕ, z) are the cylindrical coordinates, kz and kr are the transverse and longitudinal wave vector components of SPP. J ± m is the ±m-order Bessel function of the first kind. Dielectric constant of the gold film is obtained by Drude model. Based on Maxwell’s equations [14-16], we can derive the other electromagnetic field components in terms of .

The electric field components in the time domain are actually a Fourier transform of the frequency-domain ones [9,13]:

where the subscript j = x, y, z represents the three orthogonal components.

2.2

Spatiotemporal evolution of the single ultrafast plasmonic vortex

The variation of the single ultrafast plasmonic vortex with time is usually accompanied by the evolution of the beam cross-section energy flow and electric field distribution, hence Fig. 2 shows the variation of the phase, energy flow, and electric field intensity distributions of the SPP field generated at different moments of time when the incident pulse carries the initial topological charge l = 0.

Figure. 2.

The spatiotemporal evolution of the SPP field with initial l = 0 in the incident pulse under the analytical derivation and the FDTD simulation. (a1)–(a5) Evolution of the phase distribution of Ez, and the arrow representing Poynting Vector. (b1)–(b5) Evolution of the intensity distribution of the |Ez|2 field. (c)–(d) are corresponding to (a)–(b), respectively. The time range studied is from t = –200 fs to t = 200 fs. All figures are normalized to their own maximum values.

The resolved results in Figs. 2(a)-(b) show the spin-orbit transition process from σ1 = +1 to l1 = +1 in the time range from t = -200 fs to t = 0 fs (t = 0 fs corresponds to the middle of the pulse) and from σ2 = -1 to l2 = -1 in the time range from t = 0 fs to t = 200 fs. As a result, the helical phase change from clockwise to counterclockwise, which suggests that plasmonic vortices with TC l2 = -1 opposite to l1 = +1 occur during the pulse, except that at t = 0 fs, the left- and right-spinning circular polarization pulses just overlap to form a line polarization, and the vortex field disappears.

To verify the correctness of the analytical model, we simulated the pulsed SPP vortex field with the same parameters using FDTD software (Lumerical FDTD Solutions). In the FDTD simulation, the dielectric constant of the material and the dimensions of the structure are consistent with the analytical model, all the boundary conditions are set to a perfectly matched layer (PML), and the mesh size is chosen to be 40 nm in the z-direction and 25 nm in the xy-plane.The frequency-domain SPP field is recorded using a frequency-domain monitor in the xy-plane of the gold surface, and thus a time-varying SPP field can be obtained by the Fourier transform of the frequency-domain results. Figures 2(c1)-(c5) and 2(d1)-(d5) show the FDTD results of the phase, energy flow distribution and electric field strength of Ez for the case of topological charge l = 0, which are all in agreement with the analytical results, verifying the correctness of the analytical model. Therefore, in the next presentation, we show only the analytical results.

The case of the incident pulse with an initial topological charge of l = -1 corresponds to Figs. 3(a)-(b). Figs. 3(a1)-(a5) show the phase distribution of the SPP, where the central part initially shows a uniform phase distribution (Fig. 3(a1)), and then two new phase singularities with l = -1 appear in the x-direction (black circles in Fig. 3(a2)) and gradually move toward the center to form a helical phase pattern with l = -2 (Fig. 3(a5)), and at the same time, two l = +1 phase singularities (purple circles in Fig. 3(a2)) and move away from the center (Figs. 3(a2)-(a4)). For the energy flow, there is initially no a ring-shaped energy flow (Fig. 3(a1)), then two rings gradually appear in the energy flow (Fig. 3(a3)) corresponding to the two phase singularities of l = -1 (Fig. 3(a3)) and the two dark points around the center (Fig. 3(b3)), and finally the two rings merge close together to form a larger ring (Fig. 3(b5)), while the two phase singularities of l = -1 merge to form a central l = -2 singularity. As for the intensity distribution, it evolves from a bright spot (t = -200 fs) to a ring pattern (t = 200 fs). These results validate the singularity fusion process where the topological charge of the SPP field changes from l1 = σ1-1 = 0 to l2 = σ2-1 = -2 during the pulse action.

Figure. 3.

Spatiotemporal evolution of the SPP field with initial l = -1&-2 in the incident pulse. (a1)–(a5) Evolution of the phase distribution of Ez at l = 1, and the arrow representing Poynting vector. (b1)–(b5) Evolution of the intensity distribution of the |Ez|2 field at l = -1. (c)-(d) for l = -2 corresponds to (a)-(b). The time range studied is from t = -200 fs to t = 200 fs. All figures are normalized to their own maximum values.

The case of the incident pulse with an initial topological charge of l = -2 corresponds to Fig. 3(c)-(d). The patterns of change are similar, with two new phase singularities appearing in the x-direction (black circle in Fig. 3(c2)) and gradually moving toward the center, while two phase singularities with l = 1 appear in the y-direction (purple circle in Fig. 3(c2)) and move away from the center. The Poynting vector near the region where the phase singularities exist is always 0, and the initially smaller toroidal energy flow forms a larger toroidal energy flow as the TC of the phase singularities increases.

2.3

Spatiotemporal evolution of the ultrafast plasmonic vortices array

The case of the sixfold symmetric structure yields Hexagonal, Hexagonal vortex, Kagome and Honeycomb array plasmonic fields [16]. Here, we use the two orthogonally circularly polarized sub-pulses carrying the same TC to achieve the transformation of different kinds of plasmonic vortex array fields, where l = 0 is the case of helical phase switching (Fig. 4 (a)-(b)), which corresponds to the Hexagonal vortex array in which each vortex also exhibits an inverted phase. The case l = -1 (Fig. 4 (c)-(d)) is the case of phase singularity generation and corresponds to the evolution of Hexagonal to Kagome. l = -2 is the case of phase singularity with an increase in topological charge (Fig. 4 (e)-(f)) and corresponds to the evolution of Hexagonal vortex to Honeycomb.

Figure. 4.

Spatiotemporal evolution of the SPP array field with initial l = 0, –1&–2 in the incident pulse. (a1)–(a5) Evolution of the phase distribution of Ez at l = 0, and the arrow representing Poynting vector. (b1)–(b5) Evolution of the intensity distribution of the |Ez|2 field at l = 0. (c)-(d) for l = -1 corresponds to (a)-(b). (e)-(f) for l = –2 corresponds to (a)-(b).The time range studied is from t = –200 fs to t = 200 fs. All figures are normalized to their own maximum values.

For the TC = 0 case, the light intensity distribution at l1 = σ1 + 0 = +1, a complete array cell contains seven hexagonal vortices (hexagonal vortices) (Fig. 4(b1)). The phase varies from 0 to 2π and the direction of the vortex energy flow is clockwise (Fig. 4(a1)). At t = 0 fs, the polarization state of the incident light is linearly polarized, the plasmonic vortex array disappears, and a Honeycomb-like distribution is presented in the electric field intensity (Fig. 4(b3)), but the phase clearly does not show a triangular fragmentation (Fig. 4(a3)). From t = 0 fs to t = 200 fs, the vortex reappears, but the phase changes from 0 to 2π and the direction of the vortex energy flow changes to counterclockwise (Fig. 4(a4)-(a5)).

For the TC = -1 case, the light intensity distribution at l1 = σ1 -1 = 0, a complete vortex array contains seven bright spots (Hexagonal) (Fig. 4(c1)). Then, the central region gradually splits into two pairs of l = ±1 phase singularities, with the two l = -1 phase singularities moving closer to the center in the x direction and the two l = +1 phase singularities moving away from along in the y direction, while the six surrounding bright spots also form two l = -1 phase singularities in the y direction and move closer to the new cell (Fig. 4(c2)-(c4)). Eventually, a phase distribution similar to a second-order phase vortex is formed in the center, and the six next to it are similar to a first-order phase vortex, but with discontinuous phase variations along the azimuthal direction (Fig. 4(c5)), a so-called Kagome array. In the region of phase singularities, a toroidal energy flow is always formed. In the intensity distribution, the bright spot is also elongated into an elongated petal-like shape due to the shift of the singularity (Fig. 4(d5)).

For the TC = -2 case, the light intensity distribution at11=σ1+1 = -1,a complete array cell contains seven hexagonal vortices (Hexagonal vortex) (Fig. 4(f1)). Each vortex has a spiral distribution. The phase tapers counterclockwise from 0-2π (Fig. 3(e1)). This dense vortex array has orbital angular momentum per vortex and can be used for multi-particle 3D manipulation. The evolution of the light field from l1 = -1 to l2 = -3 is basically consistent with the above law, with the two l = -1 singularities in the x direction converging to the central singularity, which forms a singularity line in the process, and then changes to a singularity with l = +1(Figs. 4(e2) purple circle), which continues to converge to the central l = -1 singularity (Figs. 4(e3) black circle), and the phases form a triangular phase-crossing distribution, with π-phase jumps in the neighboring triangles (Fig. 4(e5)). Due to the formation of singularity lines, the light spots are also distributed in a six-petal pattern. Each array cell formed is honeycomb, arranged in graphene-like shape, and the light intensity is triangularly shaped at the strongest point, with six array cells (Fig. 4(f5)). From the energy flow distribution, Honeycomb and Hexagonal vortex of plasmonic field, the distribution shape is the same, but the arrangement period is different (Fig. 4(e2)-(e4)), and finally at the moment of t = 200 fs, the energy flow fades, corresponding to the defocusing of pulses (Fig. 4(e5)).These ultrafast plasmonic vortex arrays are of great potential for nonlinear optical tweezers capture of multiple particles [19,20].