Generation of time-varying orbital angular momentum beams with space-time-coding digital metasurface

Abstract. The recently proposed extreme-ultraviolet beams with time-varying orbital angular momentum (OAM) realized by high-harmonic generation provide extraordinary tools for quantum excitation control and particle manipulation. However, such an approach is not easily scalable to other frequency regimes. We design a space-time-coding digital metasurface operating in the microwave regime to experimentally generate time-varying OAM beams. Due to the flexible programmability of the metasurface, a higher-order twist in the envelope wavefront structure of time-varying OAM beams can be further designed as an additional degree of freedom. The time-varying OAM field patterns are dynamically mapped by developing a two-probe measurement technique. Our approach in combining the programmability of space-time-coding digital metasurfaces and the two-probe measurement technique provides a versatile platform for generating and observing time-varying OAM and other spatiotemporal excitations in general. The proposed time-varying OAM beams have application potentials in particle manipulation, time-division multiplexing, and information encryption.


Introduction
Electromagnetic waves carry both linear and angular momenta. Specifically, the latter can be decomposed into spin angular momentum (SAM) and orbital angular momentum (OAM). 1 Although SAM is associated with polarization, OAM is associated with the helical wavefront expð−jlθÞ in azimuthal angle θ with an integer topological charge l. 2 OAM beams with distinct topological charges are mutually orthogonal, allowing them to carry information and to be multiplexed. [3][4][5][6][7] The increased channel capacity and spectral efficiency with OAM multiplexing can thus be useful in fiber-based and free-space communications, with both prospects and challenges. 8 Moreover, OAM beams with nonzero topological charges have phase singularity with vanished intensity at the beam center, which are useful for optical trapping and lattices. [9][10][11][12] To further unlock the full potential of OAM, there have been persistent efforts in generalization of the concept. Spatiotemporal optical vortex beams [13][14][15][16][17][18][19] have been recently demonstrated with phase singularities generally in space-time coordinates, forming higher-order structures, such as a phase-singular ring. Rego et al. 20 proposed a kind of extreme-ultraviolet beam with timevarying OAM arising from high-harmonic generation. The topological charge varies in time periodically, giving a timedependent phase profile exp½−jlðtÞθ and inducing a self-torque of light. It can be potentially applied to various scenarios, including particle manipulation, [21][22][23] Bose-Einstein condensation, 24,25 and Floquet topological dynamics. 26,27 However, such a high-harmonic approach in generating time-varying OAM is not easily applicable to other frequency regimes, hindering further demonstrations. Alternatively, Sedeh et al. 28 proposed an optical metasurface with a spiral-staircase profile of modulation frequencies to implement time-varying OAM. In fact, such an approach can be feasible and generalized through a space-time-coding digital metasurface with field-programmablegate-array (FPGA) technology, particularly in the microwave regime. [29][30][31][32][33][34][35][36][37] In such a case, different metasurface responses can be digitized by controlling the bias voltages with the adoption of FPGA, which offers the metasurface powerful programmability for space-time modulation.
In this work, we experimentally construct and observe time-varying OAM beams using a space-time-coding digital metasurface in the microwave regime. We also introduce a higher-order twist in the wavefront structure of time-varying OAM beams, enabled by the flexible programmability of the metasurface. To observe the complex field pattern being varied in time, we use a two-probe measurement technique to dynamically map the time-varying OAM field pattern. A space-timecoding digital metasurface together with a two-probe fieldmapping technique provides a versatile platform to construct and observe time-varying OAM and spatiotemporal excitations in general. The proposed time-varying OAM beams can be further used in particle manipulation, time-division multiplexing, and information encryption.

Theory of Time-Varying OAM Beams Generation
To generate normal OAM beams with phase profile expð−jlθÞ by a metasurface [time harmonic convention expðjωtÞ, 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 exp½−jlðtÞθ with topological charge lðtÞ varying linearly with time, 20 manifesting a timedependent azimuthal phase gradient decided by lðtÞ. To realize such properties, we consider a space-time-coding digital metasurface with active modulation of reflection phase depending on both azimuthal angle θ and time t as Φðθ; tÞ ¼ −lðtÞθ: (1) When the lðtÞ ¼ Nt∕T is chosen to vary linearly in time with a modulation period T, we obtain N consecutive integer l from 0 to N − 1 at time instants t ¼ lT∕N. The metasurface is divided into finite N azimuthal sectors to implement the N different OAM profiles. Due to the discretization of the metasurface, the OAM phase profiles with l and l AE N are in fact equivalent to each other by assuming each sector can only give a particular reflection phase at a particular time instance, so the value of lðtÞ cannot increase to infinity but repeats from −N∕2 to N∕2 in every period T. 28 Under a monochromatic illumination E 0 expðjωtÞ at signal radial frequency ω on the metasurface with spatiotemporal phase modulation Φðθ; tÞ, the reflected field becomes E r ðρ; θ; tÞ ¼ E 0 Aðρ; θÞ expðjωtÞ exp −jN t T θ ; when the modulation frequency is much lower than the signal frequency. The Aðρ; θÞ describes the spatial dependency of the scattered field in cylindrical coordinates ðρ; θÞ, and the coefficient of θ becomes lðtÞ as Nt∕T, which denotes the time-varying topological charge of the OAM beam.
With the flexible programmability of the metasurface, the envelope wavefront structure of the above-mentioned timevarying OAM beams can be further designed. We modify the spatiotemporal phase modulation in Eq. (1) as Φ w ðθ; tÞ ¼ −lðtÞðθ − ηðtÞÞ; (3) where lðtÞ ¼ Nt∕T and ηðtÞ is defined as the time-dependent twisting angle. As an example here, we design ηðtÞ as 2πwt∕T, a linear function in time, where w is defined as the winding number of the time-varying OAM beam. Then the reflected field can be expressed as Here we consider the case of w ¼ −1, N ¼ 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 ¼ lT∕N show different OAM phase profiles, taken as l ¼ 0, 1, 2, 3, 4 ð−4Þ, −3, −2, −1 periodically. We remark that it is a superposition of two OAM modes: l ¼ 4 and l ¼ −4 at t ¼ T∕2. By "geometrically" joining up all the zero phase (cyan color) and π phase (red color) at different time instances, the wavefront structure shows up a higher-order twist for a nonzero w (w ¼ −1 in this example) compared with w ¼ 0 case (see Fig. S1 in the Supplementary Material). In other words, the envelope wavefront of the time-varying OAM beam (w ¼ 0) is further twisted with w round in one period of T. Here we only consider the integer case of w ¼ −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.
In our paper, we generate the time-varying OAM beams with w ¼ 0 and w ¼ −1, respectively. The case of w ¼ −1 is illustrated in the main text, whereas the results of w ¼ 0 can be found in Sec. 2 in the Supplementary Material for comparison. To generate the time-varying OAM beam with w ¼ −1 by a space-time-coding digital metasurface, as shown in Fig. 1(b), we divide the metasurface into N ¼ 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 exp½−jðNt∕TÞðθ − 2πwt∕TÞ in Eq. (4), with winding number w ¼ −1. For easier implementation, we only adopt eight different time frames with eight coding states in a period of T to generate the integer lðtÞ, 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 ¼ −1. The location of the zero phase takes a time of T∕jwj 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][30][31][32][33][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 m2πt∕T to the reflection phase.

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 × 20 ¼ 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 The element is composed of three copper layers printed on two substrate layers (Rogers 4003C, ε r ¼ 3.55, tan δ ¼ 0.0027) and a bonding layer (Rogers 4450F, ε r ¼ 3.52, tan δ ¼ 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.
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 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.

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 pattern of the generated time-varying OAM beams at any moment during a modulation cycle. Figure 3(b) illustrates the measured near-field amplitude and phase patterns of the time-varying OAM beams with w ¼ −1 at different instants of time in one period of T, showing the timevarying 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 ¼ 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.

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 0 ðtÞ by decomposing the field into different angular momenta, 38,39 S l 0 ðtÞ ¼ where Eðρ; θ; tÞ denotes the measured time-varying OAM field in Fig. 3(b), r 1 and r 2 are the radial range of the electric field, and l 0 is the OAM component to check on the measured field. The OAM intensity spectra for the measured field at different instants of time are shown in Fig. 4(a). Each spectrum contains different l 0 components, and it is normalized to the maximum component. At each time instant, the OAM field Eðρ; θ; tÞ has a topological charge lðtÞ ¼ Nt∕T. As can be seen, the OAM intensity spectrum at each time instant is relatively dominated by the component l 0 ¼ lðtÞ (marked by a red star), compared with the parasitic ones, showing that the generated beam contains time-varying OAM modes as the theoretical expectation shown in Fig. 1(a). Here we define OAM mode purity as PðtÞ ¼ S lðtÞ ∕ P l 0 S l 0 and obtain the measured mode purity for the dominant topological charges lðtÞ, as shown in the black bar in Fig. 4(b). On the other hand, we can also calculate the theoretical results for the mode purity, as shown in the same figure. We note that the theoretical mode purity is not 100% due to the spatial discretization of the profile exp½−jlðtÞθ into eight sectors of constant reflection phases, so that the theoretical mode purity deceases from 100% at l ¼ 0 to around 50% at l ¼ 4 ð−4Þ. Nevertheless, the experimentally extracted mode purity agrees well with the theoretical results. The difference between measured and theoretical results comes from the near-field pattern measurement error, nonuniform azimuthal intensity, and the feed-horn blocking effect. In addition, the insufficient scanning range of the field pattern also leads to the lower mode purity, especially for the l ¼ 4 ð−4Þ case.
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 ðρ; θ; tÞ 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 Eðρ; θ; tÞ using φðtÞ ¼ arg the realization of the winding number. This also indicates the winding number is a real degree of freedom of time-varying OAM and can be used for information encoding, and it can be decoded from the measured OAM field patterns. This additional degree of freedom is enabled by the flexible programmability of the space-time coding metasurface.

Discussion
The space-time-coding digital metasurfaces provide a powerful and flexible platform to manipulate the electromagnetic waves both in the spatial and temporal domains, which contributes to the realization of time-varying OAM beam generation in the current work. In fact, we have only considered the integer case of winding number w ¼ −1 as a typical example. The extension to other values, including fractional value, can also be realized with an adjustment of the space-time-coding scheme. For 0 < jwj < 1, the azimuthal position of zero phase needs to take a time of T∕jwj for a full cycle. Then the coding sequence needs to be modified with a longer period of T∕jwj. For jwj > 1, the zero phase rotates around jwj circles in one period of T. The number N of OAM modes varying in one period of T needs to be added to acquire more frames of OAM patterns to characterize the winding number w of time-varying OAM beams. For experimental verification of the time-varying OAM beam generation, one difficulty is the dynamic measurement of the time-varying amplitude and phase field pattern. The traditional microwave near-field scanning system (which conventionally features one probe) cannot fulfill our demands. The usage of a two-probe measurement technique can observe the OAM amplitude and phase pattern varying in the time domain, which enables the time-varying OAM validation in this work. We note that the S11 signal at the feed horn can also act as the second probe, serving similar purpose of the probe R. Such a two-probe technique might also be applied in other time-varying scenarios to measure other spatiotemporal excitations.
Our proposed metasurface with space-time phase modulation is implemented in the microwave regime. However, it can be extended to other frequency regimes with higher frequencies in terms of principle. From the perspective of implementation, we will need a meta-atom response that can be controlled and tunable in space and time in that frequency regime. For example, in the terahertz regime, by dynamically exciting different metamaterial structures, a reconfigurable wavefront is possible. 40 In the optical regime, a space-time varying profile of indices can be achieved by the nonlinear Kerr effect. 41 As a brief picture, a micromirror device or an spatial light modulator can be used to excite different locations on metasurface to change the local indices, e.g., using the nonlinear Kerr effect, so that the spatiotemporal profile of the OAM in this work can be achieved based on the elaborated principle. The advantages of introducing time modulation can be twofold. An additional degree of freedom in controlling OAM with a time-varying profile can be used to enhance communication capacity as an immediate application. 42 Moreover, the inhomogeneous OAM profile in both the spatial and temporal domains can also be potentially useful for obtaining unconventional particle trapping, sorting dynamics, and controlling particle trapping stability. 43 In summary, we have experimentally generated time-varying OAM beams by a space-time-coding digital metasurface in the microwave regime. A higher-order twist described by winding number w in the envelope wavefront structure of time-varying OAM beams is proposed and demonstrated. We develop a twoprobe measurement technique to dynamically map the timevarying OAM field patterns and perform the OAM spectrum analysis. The proposed time-varying OAM beams provide new routes in particle manipulation, time-division multiplexing, and information encryption.