Generation of terahertz vector beams using dielectric metasurfaces via spin-decoupled phase control

2.1 Structure design

The basic unit cell of the dielectric metasurface is schematically illustrated in Figure 2A, which is a rectangular-shape silicon pillar on the same silicon substrate. The pillar has a period of P_f = P_s = 150 μm and a height of h = 200 μm. Such a pillar has been demonstrated to function as an anisotropic effective truncated waveguide, whose phase responses of the two eigen orthogonally polarized waves can be freely controlled to individually cover the whole 2π range by changing the width D_f and length D_s [55]. This kind of phase caused by the physical dimension of the pillar is called dynamic phase. To further incorporate geometric phase, it is required to introduce rotation operation. The inset of Figure 2A illustrates a schematic of the relation of the pillar and the xy basis. The fs basis represents the local coordinate of the pillar which has a θ rotation angle with respect to the xy basis, where f and s axes are along the two symmetric axes of the pillar. Suppose the eigen transmission matrix of the pillar in the fs basis is Tfs=[Afeiφf00Aseiφs], where A_i and φ_i represent the transmission amplitude and phase shift of the i-polarized wave with i∈ {f, s}, respectively. To increase the efficiency of polarization control, we consider a special case of A_f = A_s = A and φ_s– φ_f = π, which indicates that the pillar functions as a half-wave plate. The corresponding transmission matrix in the xy basis can be calculated by applying rotation operation of coordinate transformation, which can be expressed as:

Figure 2:

Schematic of the unit silicon pillar and the simulated eigen phase and amplitude responses of the eight selected silicon pillars. (A) The basic unit cell, which is a rectangular-shape silicon pillar patterned on silicon substrate. The inset dashed box illustrates a schematic of the rotation of the unit cell. (B) Simulated eigen phase shift profiles φ_f and φ_s of the eight selected silicon pillars, as well as their phase difference profile Δφ = φ_s – φ_f, at 1.0 THz, respectively. (C) Simulated eigen amplitude transmission profiles A_f and A_s of the eight selected silicon pillars, as well as their amplitude difference profile ΔA = A_s – A_f, at 1.0 THz, respectively. Both the top insets in (B) and (C) schematically illustrate the shapes of the eight selected silicon pillars.

Equation (1) indicates the polarization orientation of the output wave can be controlled by the rotation angle θ, while the phase of this polarized wave can be controlled by the dynamic phase φ_f. To show the spin-decoupled phase control ability, transmissions under the circularly polarized incidences are also calculated. For an incident LCP plane wave of Ein=22ELeiδL[1i], the half-wave plate property will enable the output wave RCP, which can be expressed as:

Similarly, for an incident RCP plane wave of Ein=22EReiδR[1−i], the output LCP wave can be expressed as:

Here, E_L(R) and δ_L(R) are the amplitude and initial phase of the incident waves with L and R representing the LCP and RCP waves, respectively; Ф_RL and Ф_LR are the corresponding additional phase shifts resulting from the interaction with the pillar, the former and latter subscripts represent the output and incident polarization states, respectively (same hereinafter). The ±2θ is the well-known geometric phase. It is seen that phase responses of the two output circularly polarized waves can be decoupled by further controlling the dynamic phase term φ_f. From these points, we could re-understand Equation (1) that the geometric phase determines the polarization rotation, while the dynamic phase determines the phase shift of the output polarized wave relative to the incident. In particular, under the linearly polarized incidence, the output wave will also be linearly polarized but with a polarization rotation of twice of the angle formed by the pillar’s fast axis and the incident polarization owing to the half-wave-plate property. Equations (2) and (3) indicate that such pillars could be used to simultaneously and individually control the polarization and phase distributions at the interfaces by designing the superposition of the LCP and RCP output components, where the polarization distribution can be further controlled by the incident polarization state, providing a versatile platform in generating various polarization-controlled CVBs.

In order to realize full 2π phase control, eight pillars numbered 1−8 are selected from the structure database developed in our previous work, which is built at 1.0 THz [55]. As shown in the top insets of Figure 2B,C, the eight selected pillars have dimensions of D_s = 102.5, 84, 76, 83, 50, 49, 44.5, 33 μm and D_f = 50, 49, 44.5, 33, 102.5, 84, 76, 83 μm, respectively. Figure 2B illustrates the corresponding simulated dynamic phases φ_f and φ_s. From left to right, both the phases are linearly increased with a step of π/4. Figure 2C illustrates the corresponding simulated amplitude transmission (square root of the power transmittance) of the sole structured surface A_f and A_s from the silicon substrate to the air. It is seen that the values are all quite near with each other in a relatively high level around 0.85. More importantly, the phase differences Δφ = φ_s – φ_f are all nearly π and the amplitude differences ΔA = A_s – A_f are all around 0. These indicate that all the selected pillars function well as half-wave plates at 1.0 THz (see Supplementary section 1), which fulfill the above derivation. As our metasurfaces rely on superposition of the output cross-polarized LCP and RCP waves to generate CVBs, such a feature ensures the co-polarized parts have negligible influence on the metasurface performance. With these eight pillars, two metasurfaces for generating VVBs and VBBs are separately designed.

2.2 Metasurface for VVB generation

To realize the function illustrated in Figure 1A, the phase distributions Ф_RL and Ф_LR should satisfy:

respectively, where r = (x² + y²)^1/2 and α = atan2(y, x) ∈ [0, 2π] represent the radius and azimuth angle of polar coordinate, respectively. According to Equations (2) and (3), the output waves under the LCP and RCP incidences can be expressed as:

respectively. Equation (6) indicates that the transmitted beam is RCP vortex beam carrying OAM of l = +2 with initial phase δ_L remained from the incident LCP wave. Equation (7) indicates that the transmitted beam is LCP beam carrying OAM of l = 0 with initial phase δ_R remained from the incident RCP wave. Under the linearly polarized incidence, the wave can be decomposed to the LCP and RCP components with equal amplitudes where E_L = E_R = E, δ_L = −γ, and δ_R = γ, the output wave is given by:

where γ = (δ_R − δ_L)/2 represents the orientation angle of the incident polarization. Equation (8) indicates that the output wave can have inhomogeneous linear polarization and phase distributions. The output polarization rotates linearly as the azimuth angle α with orientation angle of α − γ, resulting in axially symmetric distributions controlled by γ. The output phase at an arbitrary point is equal to the corresponding azimuth angle α, corresponding to a CVB with OAM of l = 1 regardless of γ. Especially, under the x-polarized incidence (γ = 0), the output is radially polarized VVB, while under the y-polarized incidence (γ = π/2), the output is azimuthally polarized VVB.

To realize the above design, the metasurface VVB generator is designed based on Equations (4) and (5) using the eight selected pillars. In the metasurface, the pillars are arranged in a square lattice. The coordinate of each lattice point (x, y) can be transformed to the polar coordinate of (r, α), at which a pillar satisfied φ_f = α and θ = α/2 should be placed. To simplify the design, the azimuth angle is divided into eight angle regions evenly. Each region contains only one type of the pillars with φ_f, whose orientation angles are accurately determined by α. The sample was fabricated using conventional lithography, together with deep reactive ion etching on a high-resistivity silicon wafer. The microscope image of the center part of the fabricated sample is shown in Figure 3B.

Figure 3:

Schematic of the experimental setup and microscope images of the two fabricated metasurfaces. (A) The experimental setup. (B) Microscope image of the center part of the fabricated metasurface VVB generator. (C) Microscope image of the center part of the fabricated metasurface VBB generator. Both the inset scale bars represent 300 μm. Their right insets show the enlarged views of the middle nine pillars.

A scanning broadband terahertz time-domain microscopy system was used to experimentally evaluate the performance of the proposed metasurface VVB generator [26], [30], [38], [55]. Figure 3A schematically illustrates the experimental setup. The terahertz beam generated from the emitter was firstly collimated by a lens to form a quasi–Gauss beam, which was almost x-polarized. Two metallic terahertz grid polarizers and a quarter-wave plate were then employed to control the incident polarization state. Polarizer 1 was used to further enable the incident terahertz beam to be x-polarized. The LCP and RCP beams could be obtained by adjusting the quarter-wave plate without placing polarizer 2 behind it, while the x- and y-polarized beams could be obtained by further placing and adjusting polarizer 2. After the polarization of the incident terahertz beam was controlled, it normally illuminates the sample from the substrate side. A photoconductive-antenna probe was then placed above the sample to map the output terahertz beam via raster scanning. Here, the probe could only detect the x-polarized output. Under the circularly polarized incidences, the outputs were cross-circularly-polarized, so measuring the x-polarized component (E_xL and E_xR) only was enough to exhibit the performances. Under the linearly polarized incidences, however, two orthogonally polarized components should be measured. We did this by rotating the sample and polarizer 2 in combination in the measurements. For example, suppose the initial configuration was used to measure E_xx, then E_45°x, E_yx, and E_−45°x were measured by simultaneously rotating the sample and polarizer 2 by −45°, 90°, and 45°, respectively. E_yy was measured by solely rotating the sample by 90° as compared to that of E_xx, then E_45°y, E_xy, and E_−45°y were measured by simultaneously rotating the sample and polarizer 2 by −45°, 90°, and 45°, respectively. It should be noted that this measuring method cannot rebuild the polarization distributions, since the scanning grid relative to the metasurfaces before and after the rotation cannot perfect overlap with each other. However, this method allows us to show the vector and OAM properties of the generated beams in a similar demonstration way as those in the optical range [40], [43].

In the measurement, the output beam was mapped in the xy plane at a height of 5 mm above the sample. The scanning area is 6 × 6 mm with a step of 0.2 mm. Figure 4A,C illustrate the measured normalized intensity distributions |E_xL|² and |E_xR|² at 1.1 THz under the LCP and RCP incidences, while Figure 4B,D illustrate the corresponding measured phase distributions, respectively. The clear donut-shape intensity distribution and 4π phase loop indicate a good vortex beam output carrying OAM of l = +2 under the LCP incidence. The Gaussian-like intensity distribution and nearly homogeneous phase distribution indicate that no OAM is attached from the sample under the RCP incidence. Figure 4E–L illustrate the measured normalized intensity distributions |E_xx|², |E_45°x|², |E_yx|², and |E_−45°x|² under the x-polarized incidence, as well as |E_xy|², |E_45°y|², |E_yy|², and |E_−45°y|² under the y-polarized incidence at 1.1 THz, respectively. Clear “s”-shape intensity distributions are observed whose rotation is determined by the incident and detected polarizations. The rotation feature of the “s”-shape intensity distributions with regard to the detection polarizations under the x- and y-polarized incidences, as well as the azimuth angles of the maximum intensities, are consistent with previous reports [40], [43], indicating a radially and an azimuthally polarized VVBs with both OAMs of l = +1. To further verify the measured results, numerical simulations were also carried out. The insets in Figure 4 are the corresponding simulated distributions under the same conditions at 1.0 THz, which are consistent with the measured results. The efficiencies of the VVB generator are also evaluated in simulations, which are calculated by normalizing the integrated intensity of the output beam in the air side above the metasurface to that of the input beam inside the substrate. Under the LCP, RCP, x- and y-polarized incidences, the efficiencies of the corresponding desired output beams are ∼60, ∼51, ∼57, and ∼58%, respectively.

Figure 4:

Experimental and simulated performance of the metasurface VVB generator. (A, B) Measured normalized intensity distributions and phase distributions of E_xL. (C, D) Measured normalized intensity distributions and phase distributions of E_xR. (E–L) Measured normalized intensity distributions of E_xx, E_45°x, E_yx, E_−45°x, E_xy, E_45°y, E_yy, and E_−45°y, respectively. All the measured results are plotted in the xy plane at 1.1 THz. The former and latter inset arrows in the bottom indicate the input and detect polarizations, respectively. The inset figures in the top-right positions are the corresponding simulated normalized intensity and phase distributions at 1.0 THz. The diameters of the circles in (E–L) are all 6 mm.

2.3 Metasurface for VBB generation

According to the above designing strategy, this method should be able to generate Bessel beams under the circularly polarized incidences. Compared to the vortex beam, the generation of Bessel beams further requires a phase gradient along the radial direction. For an l-th order Bessel beam, the interfacial phase distribution in the polar basis can be expressed as [37]: Ф(r, α) = −2πrNA/λ + lα, where NA = λ/P is the numerical aperture, λ is the working wavelength, P is the overall period of 2π phase coverage along the radial direction, and l is also the topological charge which represents the value of the carried OAM. To realize the function illustrated in Figure 1B, the phase distributions Ф_RL and Ф_LR should satisfy:

respectively. According to Equations (2) and (3), the output waves under the LCP and RCP incidences can be expressed as:

respectively. Equation (11) indicates that the transmitted beam is RCP Bessel beam carrying OAM of l = +1 with initial phase δ_L remained from the incident LCP wave. Equation (12) indicates that the transmitted beam is LCP beam carrying OAM of l = −1 with initial phase δ_R remained from the incident RCP wave. Under the linearly polarized incidence, the output wave is given by:

It can be seen that the output beam still has Bessel phase distribution but no longer carries any OAM (l = 0). The polarization response is the same as that of the VVB generator as indicated by Equations (8) and (13). Under the x-polarized incidence (γ = 0), the output is radially polarized VBB, while under y-polarized incidence (γ = π/2), the output is azimuthally polarized VBB.

To realize the above design, the azimuth angle is also divided into eight angle regions evenly, meanwhile, we also divide the metasurface into several regions along the radial direction with a phase interval of π/4. In this case, according to Equations (9) and (10), each newly formed intersecting region contains one type of the pillars with φ_f = −2πr/P and P = 1240 μm, whose orientation angles are also accurately determined by α with θ = α/2. Figure 3C illustrates the microscope image of the center part of the fabricated sample.

To experimentally characterize the proposed metasurface VBB generator, the same method was utilized as for the VVB generator. In the measurement, the output beam was mapped in the xy plane at a height of 7 mm above the sample. The scanning area is 6 × 6 mm with a step of 0.2 mm. Figure 5A,D illustrate the measured normalized intensity distributions |E_xL|² and |E_xR|² at 1.1 THz under the LCP and RCP incidences, while Figure 5B,E illustrate the corresponding phase distributions, respectively. The Bessel distributions with donut-shape intensity profiles at the center and the spiral phase distributions with ±2π phase loops indicate that the output beams are +1-th order and −1-th order Bessel beams, respectively. To illustrate the diffraction-free feature of the Bessel beams, the intensity distributions in the yz plane under the LCP and RCP incidences were also measured, as illustrated in Figure 5C,F, respectively. The distributions are measured from −2.5 to 2.5 mm with a step of 0.1 mm along the y direction, and from 2 to 12 mm with a step of 0.5 mm along the z direction. It is seen the diffraction-free distances are longer than 20λ. Figures 5G to 5N illustrate the measured normalized intensity distributions |E_xx|², |E_45°x|², |E_yx|², and |E_−45°x|² under the x-polarized incidence, as well as |E_xy|², |E_45°y|², |E_yy|², and |E_−45°y|² under the y-polarized incidence at 1.1 THz, respectively. Clear split-lobe patterns (including center main lobe and outer sidelobes) are observed, indicating vector beams carrying OAM of l = 0. Under the x-polarized incidence, the output lobes are the strongest along the detection polarization directions, indicating a radially polarized VBB. Under the y-polarized incidence, the output lobes are the strongest along the directions perpendicular to the detection polarization directions, indicating an azimuthally polarized VBB. The insets in Figure 5 illustrate the corresponding simulated distributions at 1.0 THz, which are consistent well with the measured results. The simulated polarization distributions of the metasurface VBB generator can be seen in Supplementary section 2. Under the LCP, RCP, x- and y-polarized incidences, the simulated efficiencies of the corresponding desired output beams are ∼59, ∼59, ∼60, and ∼61%, respectively.

Figure 5:

Experimental and simulated performance of the metasurface VBB generator. (A, D, G–N) Measured normalized intensity distributions of E_xL, E_xR, E_xx, E_45°x, E_yx, E_−45°x, E_xy, E_45°y, E_yy, and E_−45°y in the xy plane, respectively. (B, E) Measured phase distributions of E_xL and E_xR in the xy plane, respectively. (C, F) Measured normalized intensity distributions of E_xL and E_xR in the yz plane, respectively. All the measured results are plotted at 1.1 THz. The former and latter inset arrows in the bottom indicate the input and detect polarizations, respectively. The inset figures in the top-right positions are the corresponding simulated normalized intensity and phase distributions at 1.0 THz. The diameters of the circles in (G–N) are all 6 mm.

It is seen from Figures 4 and 5 that the measured phase distributions are not as good as those in the simulations in the areas where the intensities tend to be zero. This can be attributed to the limited signal-to-noise ratio of our system, which cannot measure the accurate phase values in these areas. Besides, the applied 0.2 mm scanning step in the measurements also reduced the phase resolution to some extent. Nevertheless, our measured phase distributions can well demonstrate the OAM feature of the generated beams, which is an important parameter of CVBs. The working frequency of the fabricated metasurfaces (1.1 THz) here is slightly different from that in the design (1.0 THz). This can be attributed to the fabrication errors and the refractive index deviation from the design owing to the dispersive phase responses of the eight selected silicon pillars by the dynamic phase contributions (see Supplementary sections 3 and 4). Taking the overall half-wave-plate performances of the eight selected silicon pillars into account, the effective working bandwidth is around 0.04 THz. Besides, it is more interesting to note that there are two phase singularities in both the measured and simulated phase distributions of the generated vortex beam carrying OAM of l = +2 by the metasurface VVB generator under the LCP incidence, as illustrated in Figure 4B. This is caused by the phase discontinuous feature of the metasurface, where the ideal single phase singularity will be divided into two. The distance between the two phase singularities increases as the pixel size of the structure increases, and also as the number of the angle regions divided in the azimuthal direction decreases. Another reason should be the incomplete circular-polarization conversion efficiency of the composed structures, the interference effect between the output RCP and residual LCP components also contributes to the emergence of the two phase singularities since the x-polarized component was measured here. Detail information can be found in the calculated results in the Supplementary section 5. Owing to the fabrication error induced working frequency shift, the actual performance of the structures’ half-wave plate property reduces, so the two phase singularities are more obvious in the measured results.