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Here, we adopt rectangular silicon pillars and a silicon substrate to construct a transmission-type metasurface, as shown in Fig. 1 and 2a. Each silicon pillar can be regarded as a birefringent scatterer which modulates the complex amplitude of the transmitted EM waves along its long and short axes differently. When the long/short axis is along the x-/y-direction, its transmissive feature can be described as:
Fig. 2 a Schematic of a designed meta-atom. b Simulated transmission amplitude and phase spectrum of a meta-atom with L = 60 μm and W = 80 μm, under x-pol incidence. c−f Simulated transmission c, e amplitudes and d, f phases under c, d x-pol and e, f y-pol incidences with different L and W at 0.8 THz.
$$ {\mathbf{T}} = \left[ {\begin{array}{*{20}{c}} {{T_{xx}}}&0 \\ 0&{{T_{yy}}} \end{array}} \right] $$ (1) where ${T_{xx}} = {t_{xx}}{e^{i{\phi _{xx}}}}$ and ${T_{yy}} = {t_{yy}}{e^{i{\phi _{yy}}}}$ are transmission coefficients under x- and y-polarized incidence. Then the transmitted field can be given as:
$$ \left[ {\begin{array}{*{20}{c}} {E_x^{out}} \\ {E_y^{out}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{T_{xx}}}&0 \\ 0&{{T_{yy}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {E_x^{in}} \\ {E_y^{in}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{T_{xx}}E_x^{in}} \\ {{T_{yy}}E_y^{in}} \end{array}} \right] $$ (2) where there is no polarization transformation. In this way, the transmitted x- and y-pol fields can be simultaneously modulated by changing the transmissive feature of every meta-atom.
To properly suit the working frequency (0.8 THz), the parameters of meta-atoms are carefully chosen, including period p = 120 μm, height h = 400 μm, substrate thickness d = 600 μm, with length L and width W varying from 20 μm to 110 μm (see Section A, Supplementary Information). Afterward, we implement full-wave simulations using the commercial software CST Microwave Studio 2021 (see more details in the Numerical Simulation Section, Methods). A meta-atom’s transmission amplitude and phase spectrum with L = 60 μm and W = 80 μm are simulated under x-pol incidence (Fig. 2b), showing a high transmission amplitude from 0.6 THz to 1.0 THz. At the working frequency of 0.8 THz, L and W are swept from 20 μm to 110 μm to calculate the corresponding transmission amplitudes and phases under x-pol and y-pol incidence, shown in Fig. 2c–f. It can be seen when L and W change, the amplitudes remain high and stable (above 0.6 and below 0.8), while phases cover the whole 360° range for each polarization. In this way, we select 144 meta-atoms with a phase gradient of 30° to cover the phase range of 0−360° for both x-pol and y-pol incident waves. It is worth noting that the phase gradient of 30° can theoretically provide higher accuracy, while simultaneously satisfying the calculation and fabrication allowance (with more details in Section A, Supplementary Information).
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To realize the designed function, the meta-atoms need to exhibit the following phase distributions:
$$ \left\{ {\begin{array}{*{20}{c}} {\phi _1^x\left( {r,\theta } \right) = \phi _D^x\left( \theta \right) + {\phi _f}\left( r \right)} \\ {\phi _1^y\left( {r,\theta } \right) = \phi _D^y\left( \theta \right) + {\phi _f}\left( r \right)} \end{array}} \right. $$ (3) Here, (r, θ) is the in-plane position in the polar coordinate system, $\phi _D^x\left( \theta \right)$ and $\phi _D^y\left( \theta \right)$ represent the Dammann phases for x-pol and y-pol incidences, ${\phi _f}\left( r \right) = - \dfrac{{2{\text π} }}{\lambda }\left( {\sqrt {{r^2} + {f^2}} - f} \right)$ is the focusing phase to converge the transmitted waves (λ = 375 μm) into the focal plane at focus length f = 7.5 mm. Traditionally, the Dammann phase is the phase profile for Dammann gratings, a kind of binary optics element, to generate an equal-energy field distribution among all desired diffraction orders34. In this work, we use the azimuthal Dammann phases to generate OAM combs, while their binary phase values (0 or π) are decided using azimuthal transition points, which can be derived from a numerical optimization method35. These transition points are crucial to generate OAM combs. Then, the transmission coefficient of the first designed azimuthal Dammann phase mask can be described as:
$$ {T_1}\left( \theta \right) = \exp \left( {i{\phi _D}\left( \theta \right)} \right) = \sum\limits_{n = - \infty }^{ + \infty } {{C_n}\exp \left( {{{i2{\text π} \theta n} \mathord{\left/ {\vphantom {{i2\pi \theta n} \Lambda }} \right. } \Lambda }} \right)} $$ (4) where Λ is the azimuthal period, and Cn is the coefficient of the nth OAM mode, the value of which can be determined by azimuthal transition points. Thus, modulated by the azimuthal Dammann phase in the θ-space (Fig. 3), OAM combs that contain several OAM series with different topological charges l, are formed after EM waves pass through the metasurfaces. The focusing phase is also added to observe OAM combs on the focal plane. Theoretically, the process of OAM mode analysis can be described by the Fourier transform in the polar coordinate system, and then the OAM spectrum of the field on the focal plane can be written as:
Fig. 3 a, b Schematic illustration of the process for creating phase profiles of sample 1 under a x-pol and b y-pol incidence.
$$ {E_1}\left( l \right) = \mathcal{F}\left\{ {{T_1}\left( \theta \right)} \right\} = \sum\limits_{n = - \infty }^{ + \infty } {{C_n}\delta \left( {l - {{2{\text π} n} \mathord{\left/ {\vphantom {{2\pi n} \Lambda }} \right. } \Lambda }} \right)} $$ (5) where l is the topological charge of an OAM mode, $\mathcal{F}\left\{ \cdot \right\}$ denotes the Fourier transform, and $\delta \left( \cdot \right)$ represents the unit-impulse function. It is worth noting that l and θ are a pair of Fourier transform bases. In this way, the OAM comb can be realized on the focal plane by selecting a series of transition points, which determine the values of different Cn. To be specific, in Fig. 3a, the transition points are θn = {0, 0.23191, 0.42520, 0.52571, 1} × 2π to generate an OAM comb with l = {–3, –2, …, +3}. By contrast, in Fig. 3b, the transition points are θn = {0, 0.18240, 0.27424, 0.58581, 0.67967, 0.71917, 0.82217, 0.90642, 1} × 2π to generate an OAM comb with l = {–7, –6, …, +7}. In the two cases, Λ = 2π.
Then we fabricate the first sample through the deep reactive ion etching (DRIE) process (see more details in the Sample Fabrication Section, Methods), the photo of which is demonstrated in Fig. 4a. Sample 1 contains 80 × 80 rectangular silicon pillars as meta-atoms with a total size of 9.6 mm × 9.6 mm (with more details in Section B, Supplementary Information). Next, a THz near-field scanning system is adopted to characterize the function of sample 1, as schematically shown in Fig. 4b. The x-pol (or y-pol) incident waves normally shine sample 1, and then the field distributions of the corresponding polarization are detected by a probe on the focal plane with z = 7.5 mm (see more details in the Experimental Setup Section, Methods). Fig. 4c and d show the simulated (in the 1st row) and the measured (in the 2nd row) x-pol (or y-pol) electric field distribution of intensity (|Ex|2 or |Ey|2) and phase (ϕx or ϕy) under x-pol (or y-pol) incidence on the focal plane, suggesting that the simulated results agree with experimentally measured ones. The beams are focused on the central area, where some dark lines suggest an abrupt phase change of π. The field patterns are complicated, indicating there are multi-mode OAM waves (i.e., OAM combs). Theoretically, the electric field distributions on the focal plane (in the polar coordinate system) can be denoted by the superposition of vortex harmonics $\exp \left( {il\theta } \right)$:
Fig. 4 a A photo of part of the fabricated sample 1. b Schematic of the THz field measurement process. c, d Simulated (in the 1st row) and measured (in the 2nd row) field distributions of intensity and phase under c x-pol and d y-pol incidence on the focal plane of sample 1. e, f Calculated OAM spectrum by analyzing simulated and measured field distributions on the focal plane of sample 1 under e x-pol and f y-pol incidence.
$$ E\left( {r,\theta } \right) = \frac{1}{{\sqrt {2{\text π} } }}\sum\limits_{l = - \infty }^{ + \infty } {{a_l}\left( r \right)\exp \left( {il\theta } \right)} $$ (6) where the complex weights ${a_l}\left( r \right)$ can be calculated by:
$$ {a_l}\left( r \right) = \frac{1}{{\sqrt {2{\text π} } }}\int_0^{2{\text π} } {E\left( {r,\theta } \right)\exp \left( { - il\theta } \right)} d\theta $$ (7) Then the intensity of OAM modes l can be described as:
$$ {A_l} = \int_0^\infty {{{\left| {{a_l}\left( r \right)} \right|}^2}r} dr $$ (8) Using Eq. 7 and 8, We can analyze the simulated and measured field distributions $E\left( {r,\theta } \right)$ at each location $\left( {r,\theta } \right)$ in the polar coordinate system. Then the normalized calculated mode intensities ${P_l} = {{{A_l}} \mathord{\left/ {\vphantom {{{A_l}} {\displaystyle\sum\limits_{l = - \infty }^{ + \infty } {{A_l}} }}} \right. } {\displaystyle\sum\limits_{l = - \infty }^{ + \infty } {{A_l}} }}$ under x-pol (or y-pol) incidence are depicted in Fig. 4e and f. As theoretically predicted, these fields are the superposition of a series of OAM modes with equal intensities (i.e., OAM combs). The x-pol (or y-pol) incident waves passing through sample 1 can be converted to an OAM comb with l from –3 to +3 (or from –7 to +7), whose length is 7 (or 15). As a result, a pair of polarization-multiplexed OAM combs with different lengths are achieved. The experimental results agree with the simulation. However, there is still some deviation between them. We attribute the causes to the limit of the fabrication and measurement processes, with more details in Section C, Supplementary Information.
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Similarly, the meta-atoms for the second function need to possess the following phase distributions:
$$ \left\{ {\begin{array}{*{20}{c}} {\phi _2^x\left( {r,\theta } \right) = \phi _D^x\left( \theta \right) + \phi _v^x\left( \theta \right) + {\phi _f}\left( r \right)} \\ {\phi _2^y\left( {r,\theta } \right) = \phi _D^y\left( \theta \right) + \phi _v^y\left( \theta \right) + {\phi _f}\left( r \right)} \end{array}} \right. $$ (9) where $\phi _v^{x,y}\left( \theta \right) = l_0^{x,y}\theta $ denotes the vortex phase with a topological charge l0 for x-pol or y-pol incidence. Specifically, $l_0^x = + 3$ and $l_0^y = + 2$. Compared with the last section, the vortex phase is added to the total phase distribution. Then the transmission coefficient of the second designed azimuthal phase mask can be written as:
$$\begin{split} {T_2}\left( \theta \right) =\;& \exp \left( {i\left( {{\phi _D}\left( \theta \right) + {\phi _v}\left( \theta \right)} \right)} \right) \\=\;& \exp \left( {i{l_0}\theta } \right)\sum\limits_{n = - \infty }^{ + \infty } {{C_n}\exp \left( {{{i2{\text π} \theta n} \mathord{\left/ {\vphantom {{i2{\text π} \theta n} \Lambda }} \right. } \Lambda }} \right)} \end{split}$$ (10) After incorporating the focusing phase, on the focal plane, the OAM spectrum of the field can be analyzed by:
$$ {E_2}\left( l \right) = \mathcal{F}\left\{ {{T_2}\left( \theta \right)} \right\} = \sum\limits_{n = - \infty }^{ + \infty } {{C_n}\delta \left( {l - {{2{\text π} n} \mathord{\left/ {\vphantom {{2{\text π} n} {\Lambda - {l_0}}}} \right. } {\Lambda - {l_0}}}} \right)} $$ (11) In this way, the OAM comb can be formed on the focal plane by selecting l0, Λ, and a series of transition points. To be specific, in Fig. 5a, l0 = +3, Λ = π, and the transition points are θn = {0, 0.11596, 0.21260, 0.26286, 0.50000, 0.61596, 0.71260, 0.76286, 1} × 2π to generate an OAM comb with l = {–3, –1, +1, +3, +5, +7, +9}. By contrast, in Fig. 5b, l0 = +2, Λ = 2π/3, and the transition points are θn = {0, 0.07730, 0.14173, 0.17524, 0.33333, 0.41064, 0.47507, 0.50857, 0.66667, 0.74397, 0.80840, 0.84190, 1} × 2π to generate an OAM comb with l = {–7, –4, –1, +2, +5, +8, +11}. It can be easily seen that in the OAM spectrum, the location is l0, while the interval is 2π/Λ.
Fig. 5 a, b Schematic illustration of the process for creating phase profiles of sample 2 under a x-pol and b y-pol incidence. c, d Simulated (in the 1st row) and measured (in the 2nd row) field distributions of intensity and phase under c x-pol and d y-pol incidence on the focal plane of sample 2. e A photo of part of the fabricated sample 2. f, g Calculated OAM spectrum by analyzing simulated and measured field distributions on the focal plane of sample 2 under f x-pol and g y-pol incidence.
We then fabricate the second sample using the DRIE process, the photo of which is shown in Fig. 5e, with the same size as sample 1 (see more details in Section B, Supplementary Information). Next, the function of sample 2 is experimentally measured using the same measurement process shown in the last section. Fig. 5c and d demonstrate the simulated (in the 1st row) and the measured (in the 2nd row) x-pol (or y-pol) electric field distribution of intensity (|Ex|2 or |Ey|2) and phase (ϕx or ϕy) under x-pol (or y-pol) incidence on the focal plane, suggesting a good agreement between simulated results and experimentally measured ones. After analyzing the field distributions, the normalized calculated mode intensities under x-pol and y-pol incidence are shown in Fig. 5f and g. As theoretically predicted, these fields are OAM combs. The x-pol (or y-pol) incident waves passing through sample 2 can be converted to an OAM comb with l = {–3, –1, +1, +3, +5, +7, +9} (or l = {–7, –4, –1, +2, +5, +8, +11}). Obviously, in the OAM spectrum, the location of the x-pol (or y-pol) OAM comb is +3 (or +2), while the interval is 2 (or 3). As a result, a pair of polarization-multiplexed OAM combs with different locations and intervals are achieved. The reasons for the deviation between measured and simulated results can be found in Section C, Supplementary Information.
Generation of polarization-multiplexed terahertz orbital angular momentum combs via all-silicon metasurfaces
- Light: Advanced Manufacturing 5, Article number: (2024)
- Received: 27 February 2024
- Revised: 15 June 2024
- Accepted: 29 June 2024 Published online: 29 July 2024
doi: https://doi.org/10.37188/lam.2024.038
Abstract: Electromagnetic waves carrying orbital angular momentum (OAM), namely OAM beams, are important in various fields including optics, communications, and quantum information. However, most current schemes can only generate single or several simple OAM modes. Multi-mode OAM beams are rarely seen. This paper proposes a scheme to design metasurfaces that can generate multiple polarization-multiplexed OAM modes with equal intervals and intensities (i.e., OAM combs) working in the terahertz (THz) range. As a proof of concept, we first design a metasurface to generate a pair of polarization-multiplexed OAM combs with arbitrary mode numbers. Furthermore, another metasurface is proposed to realize a pair of polarization-multiplexed OAM combs with arbitrary locations and intervals in the OAM spectrum. Experimental results agree well with full-wave simulations, verifying a great performance of OAM combs generation. Our method may provide a new solution to designing high-capacity THz devices used in multi-mode communication systems.
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