2.1.

Theory of Time-Varying OAM Beams Generation

To generate normal OAM beams with phase profile $\exp (-jl\theta )$ by a metasurface [time harmonic convention $\exp (j\omega t)]$, the direct way is to create an azimuthal phase gradient of the scattered field with topological charge $l$, while for the time-varying OAM beams, the phase profile turns to be $\exp [-jl(t)\theta ]$ with topological charge $l(t)$ varying linearly with time,²⁰ manifesting a time-dependent azimuthal phase gradient decided by $l(t)$. To realize such properties, we consider a space-time-coding digital metasurface with active modulation of reflection phase depending on both azimuthal angle $\theta$ and time $t$ as

With the flexible programmability of the metasurface, the envelope wavefront structure of the above-mentioned time-varying OAM beams can be further designed. We modify the spatiotemporal phase modulation in Eq. (1) as

Here we consider the case of $w=-1$, $N=8$ for illustration purposes. As shown in Fig. 1(a), the envelope wavefront structure of the reflected beam has been plotted, where the constant phase contour is only about the slowly varying modulation phase envelope, and the much faster phase variation at signal frequency is omitted in the drawing for brevity. As can be seen, different time layers at $t=lT/N$ show different OAM phase profiles, taken as $l=0$, 1, 2, 3, $4(-4)$, $-3$, $-2$, $-1$ periodically. We remark that it is a superposition of two OAM modes: $l=4$ and $l=-4$ at $t=T/2$. By “geometrically” joining up all the zero phase (cyan color) and $\pi$ phase (red color) at different time instances, the wavefront structure shows up a higher-order twist for a nonzero $w$ ($w=-1$ in this example) compared with $w=0$ case (see Fig. S1 in the Supplementary Material). In other words, the envelope wavefront of the time-varying OAM beam ($w=0$) is further twisted with $w$ round in one period of $T$. Here we only consider the integer case of $w=-1$ as an example, while the $w$ can be other integers and fractional values (see Sec. 4 for details). The winding number $w$ can be seen as an additional degree of freedom in constructing the time-varying OAM beams. More information can be encoded by designing the $w$ to increase the channel capacity for data transmission. And we provide an example of winding number implementation and measurement with experimental demonstrations, as shown in Sec. 3.2.

Fig. 1

Time-varying OAM beam generation. (a) Time-varying OAM beam with topological charge $l(t)$ varying from 0, 1, 2, 3, $4(-4)$, $-3$, $-2$, and $-1$ periodically in time, with envelope phase profile [omitting $\exp (j\omega t)$ for brevity] shown at different layers in time. The cyan and red curved surfaces geometrically join up the 0 and $\pi$ phase on different layers, showing a twist completing one cycle in the clockwise direction defined with winding number $w=-1$. The beam is generated by a space-time-coding digital metasurface controlled with FPGA. (b) Digital coding scheme of the metasurface with 3-bit coding digits “1” to “8” representing the 0 deg to 315 deg in every 45 deg for the reflection phase profile (in addition to reflection from a perfect metal) at each time instance. The black stars mark the twisted trajectory of zero phase position with coding digit “1.”

In our paper, we generate the time-varying OAM beams with $w=0$ and $w=-1$, respectively. The case of $w=-1$ is illustrated in the main text, whereas the results of $w=0$ can be found in Sec. 2 in the Supplementary Material for comparison. To generate the time-varying OAM beam with $w=-1$ by a space-time-coding digital metasurface, as shown in Fig. 1(b), we divide the metasurface into $N=8$ azimuthal sectors, and each sector is dynamically controlled with 3-bit coding digits “1” to “8,” representing the reflection phase from 0 deg to 315 deg in every 45 deg (in addition to reflection from a perfect metal). The space-time-coding scheme is designed according to the phase term $\exp [-j(Nt/T)(\theta -2\pi wt/T)]$ in Eq. (4), with winding number $w=-1$. For easier implementation, we only adopt eight different time frames with eight coding states in a period of $T$ to generate the integer $l(t)$, whereas the fractional OAM states are not considered for the implementation. The details for the space-time modulation scheme can be found in Tables S1 and S2 in the Supplementary Material. In Fig. 1(b), the corresponding phase states represented by different colors show an $l$-related distribution at each time layer, and the black stars mark the azimuthal positions of zero phase (coding digit “1”) showing the higher-order twist in the envelope wavefront structure of time-varying OAM with $w=-1$. The location of the zero phase takes a time of $T/|w|$ for one revolution. Here we emphasize that the upper limit of the modulation frequency of the space-time-coding digital metasurface is typically within the megahertz range,^29–34 which is much slower than the signal frequency (gigahertz) in the microwave regime. Therefore, the modulation mechanisms and the operation only focus on the modulation envelope rather than the carrier signal. In our current work, the time-varying control is achieved through adiabatic field control at the fundamental frequency. It is also possible to instruct the output beam to another $m$’th harmonic by adding an additional phase $m2\pi t/T$ to the reflection phase.

2.2.

Space-Time-Coding Digital Metasurface

To demonstrate the above-illustrated concept, as shown in Fig. 2(a), we design and fabricate a space-time-coding digital metasurface with $20\times 20=400$ elements operating at 11 GHz. Each element is loaded with a varactor diode, whose capacitance changes with the bias voltage, leading to the frequency shift of the dipole resonance of the metasurface so that the reflection phase can be varied in time. We note that we apply the commonly used element structure embedded with a varactor diode for the realization of digital-coding metasurface as previous work,^30,31,33 but now with a biasing network design appropriate for generating time-varying OAM in Fig. S3 in the Supplementary Material and the digital coding scheme in space-time domain in Fig. 1(b). The 3D structure of each element of the proposed metasurface is illustrated in Fig. 2(b). The geometrical parameters of the element are chosen as $h_1=0.813\mathrm{mm}$, $h_2=0.1\mathrm{mm}$, $h_3=0.305\mathrm{mm}$, ${Ø}_1=1.2\mathrm{mm}$, ${Ø}_2=0.8\mathrm{mm}$, $s=0.6\mathrm{mm}$, $a=10\mathrm{mm}$, $b=5\mathrm{mm}$, $c=4\mathrm{mm}$, and $g=1\mathrm{mm}$. The element is composed of three copper layers printed on two substrate layers (Rogers 4003C, ${\epsilon }_r=3.55$, $\tan \delta =0.0027$) and a bonding layer (Rogers 4450F, ${\epsilon }_r=3.52$, $\tan \delta =0.004$). On the top layer, two rectangular metal patches are connected by a varactor diode (MAVR-000120-14110P) soldered on the metasurface by the surface mounting technology (SMT). One patch is connected to the middle layer through metallic via −. Another patch is connected to the bottom layer through via +, which is electrically isolated from the middle layer by a hollowed ring. The middle layer connects to negative “−” electrode and the bottom layer connects to positive “+” electrode, providing the bias voltage to the varactor diode through the two metallic vias.

Fig. 2

Metasurface design and reflection responses. (a) Photo of the fabricated space-time-coding digital metasurface. (b) Detailed geometrical parameters of the element structure. (c), (d) Simulated reflection amplitude and phase of the metasurface at different bias voltages, where the vertical orange line indicates the operating frequency at 11 GHz. (e), (f) Measured reflection amplitude and phase of the metasurface at different bias voltages.

To characterize the reflection responses of the metasurface, full-wave simulations are performed (CST Microwave Studio). The simulated reflection amplitude and phase at eight bias voltages from U1 to U8 are shown in Figs. 2(c) and 2(d). As can be seen, we acquire eight different reflection phases at 11 GHz with a 45-deg gradient covering a range of 315 deg, which can be used as 3-bit coding digits from “1” to “8.” The corresponding amplitudes at 11 GHz are not uniform, and their differences are controlled within $-1.7\mathrm{dB}$. The experimental results agree well with the simulation results, as shown in Figs. 2(e) and 2(f), which are obtained by measuring the reflection coefficient of the metasurface in setting the bias voltages for all the elements to be the same one. In addition, we simulate the near-field patterns of the time-varying OAM beam at different time instances, which can be found in Fig. S2 in the Supplementary Material.

3.1.

Dynamic Field Pattern Measurement of the Time-Varying OAM Beams

For experimental demonstration of the time-varying OAM beams generated by the space-time-coding digital metasurface, we directly measure the time-varying field pattern, including amplitude and phase information at various instants of time. A vector network analyzer can be used in the continuous wave (CW) time sweep mode to measure time-domain $S$-parameters. However, as the measured $S$-parameters in the time domain can start at any instant within one modulation cycle, the fields scanned at different positions cannot be compared with each other. Here in addition to the scanning probe S, we add a reference probe R whose position is fixed, and the two probes are measured synchronously. Then the results from the probe R do not vary against different probe S positions except for a time shift, which can now be used to align the $S$-parameters between signals probed at different positions (see the schematic in Figs. S4 and S5 in the Supplementary Material for more details). In our experiment, as shown in Fig. 3(a), the space-time-coding digital metasurface is applied with 3-bit digital coding sequence [Fig. 1(b)], to generate the time-varying OAM beam with a modulation period around $T=1\mathrm{ms}$ with winding number $w=-1$. The feed horn is placed 0.2 m away from the reflective metasurface to give a monochromatic excitation signal at 11 GHz polarized in the $y$ direction. Note that the feed horn is located at the beam null of the reflected OAM beams to minimize the block effect. Two open-ended waveguide probes (WR-90) are placed 0.44 m away from the metasurface. The stimulus condition of the vector network analyzer (Agilent N5230A) is set as CW time sweep mode at 11 GHz, with a sweep time of 10 ms. Then a time-domain $S$-parameter waveform (complex value) with a temporal resolution of $12.5\mu \mathrm{s}$ can be measured. By simultaneously measuring the probe R and probe S in the scanning range of $0.6\mathrm{m}\times 0.6\mathrm{m}$, with a step size of 13.6 mm, $45\times 45=2025$ groups of data are obtained. By performing data postprocessing for the received signals at all scanning positions (see details in Sec. 4 in the Supplementary Material), we can map the dynamic amplitude and phase field pattern of the generated time-varying OAM beams at any moment during a modulation cycle.

Fig. 3

Dynamic measurement of time-varying OAM field pattern ($w=-1$) using the two-probe technique. (a) Experimental scenario of dynamic electric field pattern measurement of the generated time-varying OAM beams. The probe S is movable to scan the field, and the reference probe R is fixed for synchronization processing. (b) Measured near-field amplitude and phase patterns of the time-varying OAM beams at different time instances in one period of $T$. The field patterns consisted of $45\times 45=2025\text{pixels}$ with a range of $0.6\mathrm{m}\times 0.6\mathrm{m}$.

Figure 3(b) illustrates the measured near-field amplitude and phase patterns of the time-varying OAM beams with $w=-1$ at different instants of time in one period of $T$, showing the time-varying topological charge $l$ in consecutive integer values. The amplitude patterns with nonzero $l$ show a dark area in the center due to the phase singularity along the OAM beam axis. The nonuniform intensity of the OAM along the azimuthal direction is caused by the differences in reflection amplitude for different coding states at 11 GHz, as shown in Fig. 2(e). In addition, for these phase patterns, a multiannular structure can be observed because the incident beam is seen as a spherical wave when the feed horn is placed not far away enough from the metasurface. The phase pattern of $l=0$ shows an almost homogeneous phase in the central area and the other patterns with nonzero $l$ show different spiral fringes determined by the value of $l$. The change of OAM is clearly observable as the spiral fringes vary in time. We observe that these spiral fringes are disconnected from the peripheral phase profile due to the finite size of the fabricated metasurface. Some noise points can be found in the measured field pattern, which is caused by the synchronization error when using two-probe dynamic field mapping.

3.2.

OAM Spectrum Analysis and Winding Number Decoding

To further evaluate the OAM mode purity and decode the winding number $w$ of the generated time-varying OAM beams, first we calculate the OAM intensity spectrum $S_{l^{\prime }}(t)$ by decomposing the field into different angular momenta,^38,39

Fig. 4

Spectrum analysis of time-varying OAM. (a) OAM intensity and phase spectrum of the measured time-varying OAM beams at different instants of time. (b) Measured and theoretical OAM mode purity for the dominant $l(t)$. (c) Comparison of measured and theoretical twisting phases for different time instants of dominant $l(t)$.

Next, we investigate the winding (or twisting) wavefront structure of the time-varying OAM beam. We numerically obtain the field arriving at the measurement plane for a metasurface with zero winding structure using the Fresnel diffraction formula and denote it as $E_b(\rho ,\theta ,t)$ at different time instants (see Sec. 1 in the Supplementary Material for details). Then the additional phase introduced by the twisting can be obtained from the measured field $E(\rho ,\theta ,t)$ using