Introduction

Frequency is a fundamental property of electromagnetic (EM) waves. Flexible frequency control is crucial for diverse fields ranging from microwave photonics1,2,3, wireless communications4,5,6, to optical quantum computing7. For instance, one can use optical frequency combs with equally spaced spectral lines for timing and synchronizing atomic clocks8, calibration of astronomic spectrographs9, optical arbitrary waveform generation10, and signal processing11. The ability to generate and independently manipulate multiple frequency orders is a prerequisite to achieving full utilization of spectrum resources.

Different architectures are under active investigation to implement frequency control. One common approach relies on the optical nonlinearity of bulk media. Various nonlinear phenomena, such as second/third harmonic generation12,13, sum/difference frequency generation14, four-wave mixing15,16, and cross-phase modulation17, have been demonstrated using all-optical wave-mixing. Although significant frequency shifts can be achieved, the bulk media are incompatible with photonic integration and require strict phase-matching conditions as well as high-power optical pumps. Moreover, the nonlinear process generates parasitic nonlinear processes, requiring stringent spectral and spatial filtering. Another popular approach is based on electro-optic (EO) modulators, which can directly interface electrical and optical signals. For unidirectional frequency conversion, various EO architectures have been proposed to suppress undesired symmetric sidebands, such as in-phase and quadrature (IQ) modulators using destructive interference18,19, Serrodyne modulation with a saw-tooth waveform20, two-resonator frequency shifters via the coupling between the discrete and continuum optical modes21. More complicated frequency comb generation can be implemented by cascading multiple EO phase and amplitude modulators22, placing the EO phase modulator inside a Fabry-Pérot resonator through χ(2) process23,24. Unfortunately, these EO solutions for single-/multi-frequency generation and manipulation generally require multiple modulators with complicated structures and lack the flexibility and tunability for each frequency comb line.

On the other hand, metasurfaces have transformed the field of photonics and electronics by offering exceptional control over electromagnetic waves. These 2D structures provide an attractive solution to overcome the limitations and challenges faced in frequency control. Nonlinear optical metasurfaces have emerged as a promising platform to enhance the effective nonlinearities of natural materials and relax the phase-matching requirements due to their subwavelength thickness25. The overall frequency conversion efficiency in optical bands remains low, with plasmonic structures achieving only ~10−1026,27,28 and all-dielectric designs reaching only the order of 10−629,30,31,32. In microwave bands, lumped elements such as varactor diodes33,34,35, field effect transistors36, active frequency multipliers37, and mixers38 can be incorporated into meta-atoms to boost the nonlinear responses. For example, the metasurface integrating active multiplier integrated circuits can achieve second harmonic generation with a high conversion efficiency of 85.11%37. As another promising solution, spatiotemporally modulated metasurfaces (STMMs) utilize time as another degree of freedom in metasurface design, enabling precise regulation of both frequency and momentum contents of EM waves39,40,41,42,43,44,45,46,47. Various exotic wave manipulations by STMMs have been reported, such as breaking Lorentz reciprocity constraints48,49,50, Doppler cloaks51,52,53, and direct signal modulation54,55,56. However, current nonlinear metasurfaces and STMMs incur very limited spectral and wave properties controllability. Specifically, the harmonics generated by STMMs are generally coupled to each other, limiting most STMMs to engineering only one specific harmonic. Although asynchronous STMMs can achieve multi-harmonic control by dividing the whole metasurface aperture into multiple sections57, they incur reduced aperture efficiency. Furthermore, space-fed STMMs generally result in undesired harmonics radiations in free space that pollute useful signals.

Recently, waveguide-integrated STMMs, namely, space-time-coding metasurface antennas (STCMAs), have been proposed to bridge the transformation gap between the guided wave and free-space wave41,58,59. A spatial multiplexing method is developed to generate multiple harmonic frequencies for STCMAs59. Nevertheless, it is difficult for the lattice size in each sub-metasurface satisfying the spatial Nyquist sampling requirement to avoid high-order diffraction (see Supplementary Note S3), imposing a constraint for the maximum available number of harmonics. Specifically, only three harmonics generation and independent control are demonstrated for STCMAs59. A flexible device capable of simultaneously generating and independently handling unlimited harmonic orders is still missing.

This article reports a synthetic moving envelope solution for metasurface antennas to simultaneously generate and independently control arbitrary harmonic orders. We first generalize the static envelope, reconfigurable envelope into the moving envelope that enables unidirectional frequency conversion and radiation. To realize arbitrary harmonic order synthesis, we develop a synthetic moving envelope approach that superposes multiple fundamental sinusoidal moving envelopes; each fundamental envelope corresponds to one specific harmonic conversion and radiation. The synthetic moving envelope can be implemented by simply loading the corresponding 1-bit ‘0/1’ digital STC film to the STCMA. Through several illustrative examples, we demonstrate the ability of the synthetic metasurface antenna to generate arbitrary target harmonic orders and independently control their wave behaviours and power distributions at will. Notably, the sideband-free nature of the waveguide-integrated metasurface antenna ensures that undesired harmonics are highly suppressed in free space. We design, fabricate, and experimentally test a metasurface antenna loaded with positive-intrinsic-negative (PIN) diodes to verify the proposed synthetic approach and metasurface antenna design at microwave frequencies. We also demonstrated the synthetic metasurface antenna for frequency-division multiplexing (FDM) wireless communications.

Results

Single frequency conversion

Figure 1a illustrates the conceptual illustration of the synthetic metasurface antenna for flexible frequency synthesis. In analogy to a movie projector projecting films over a screen, the synthetic metasurface antenna can map the fictitious STC film to the radiation wave in the momentum and energy (\(k-\omega \)) space. The metasurface antenna consists of an array of slot-opening meta-atoms residing on the top of the substrate-integrated waveguide (SIW) to couple guided waves into free-space propagating waves60. A continuous-wave (CW) input with an angular frequency \({\omega }_{0}=2\pi \times 22.5\) GHz is fed into the waveguide, and arbitrary target harmonic orders can be generated and radiated into free space through 1-bit spatiotemporally ON-OFF switching the meta-atoms between the radiating (coding state ‘1’) and non-radiating (coding state ‘0’) states. We design and fabricate a metasurface antenna prototype (Fig. 1b), with detailed geometric parameters of the meta-atom given in Fig. 1c. Active components in the form of PIN diodes controlled by a field-programable gate array (FPGA) are utilized to switch the radiating state of meta-atoms independently.

Fig. 1: Synthetic moving-envelope metasurface antenna for arbitrary harmonic synthesis.
figure 1

a By loading fictitious STC films, the STCMA resembles a ‘projector’ to forming synthetic momentum and frequency contents in the \(k-\omega \) space. The metasurface antenna is capable of converting the in-plane monochromatic guided waves into out-of-plane free-space waves with arbitrary, independently controllable harmonic frequencies. Each meta-atom of the STCMA incorporates PIN diodes to switch the element between radiating (‘1’) and non-radiating (‘0’) states, forming a synthetic moving envelope along the length of the waveguide. The excitation states of the meta-atoms are switched according to the applied 1-bit digital ‘0/1’ STC film from an FPGA. In this illustrated example, the guided mode is converted into free-space modes with a desired frequency comb. b Photograph of the fabricated metasurface antenna prototype. c Front and cross-section views of the meta-atom. All geometrical parameters are in mm.

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Previous studies59,61 have demonstrated that a static sinusoidal amplitude envelope along the length of the waveguiding structure results in high-directivity radiation at the same frequency as the input wave (Fig. 2a). The static sinusoidal envelope can be implemented by leveraging the time-average effect of the spatiotemporal modulation at the fundamental frequency59. However, the radiation beam direction of the metasurface antenna with a static envelope is fixed upon fabrication. The static envelope is developed to a reconfigurable or dynamic one59,60 (Fig. 2b) such that the envelope can be changed from one state to another state for wave properties control, such as beam switching or direct information modulation. However, the envelope is still fixed in time and is changed by the control system only to switch the functionalities whenever needed. For example, the equivalent envelope is kept fixed after switching the main beam direction from θ1 to θ2 (Fig. 2b). In this work, we extend the concept to the moving envelope (Fig. 2c), indicating the envelope is continuously moving in the backward or forward direction along the length of the waveguide. The moving envelope introduces an equivalent Doppler effect to the radiation beam for frequency conversion. Combining the moving envelope concept and the superposition principle, we further develop a synthetic moving envelope solution (Fig. 2d) that facilitates arbitrary harmonic order generation and their wave behaviours independent manipulation for metasurface antennas.

Fig. 2: Evolution of the synthetic moving envelope concept.
figure 2

a Static envelope results in fixed properties of the radiation waves. b Reconfigurable/dynamic envelope is capable of changing or switching properties of the radiated waves, such as the beam direction. Nevertheless, the envelope is fixed in time, and is changed only to switch the functionalities whenever needed. c Moving envelope with a continuously moving sinusoidal envelope propagating along the length of the waveguide results in unidirectional frequency conversion. d Synthetic moving envelope is the superposition of multiple basic continuously moving sinusoidal envelopes. Synthetic moving envelope facilitates the generation of arbitrary harmonic orders and independent control of their wave properties.

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We demonstrate our metasurface antenna’s arbitrary harmonic order synthesis capability starting with the unidirectional frequency conversion, i.e., translating the monochromatic input wave at one frequency into the out-of-plane propagating wave at another frequency in free space (see Fig. 3a). The sinusoidal moving envelope propagating along the x-axis takes the form of

$$E(x,\, t)={A}_{E}[1+{M}_{E}\cos \left({\Omega }_{E}t-{k}_{E}x\right)]$$
(1)

where \({A}_{E}\) is a constant amplitude and \({M}_{E}\in [{\mathrm{0,1}}]\) is the modulation depth of the amplitude modulation. \({\Omega }_{E}\) and \({k}_{E}\) are the angular temporal and spatial frequencies of the moving envelope, respectively. Taking the time derivative of a constant phase point of the moving envelope yields the velocity:

$${v}_{E}=\frac{{\Omega }_{E}}{{k}_{E}}$$
(2)
Fig. 3: STCMA unidirectional frequency conversion.
figure 3

a Schematic illustration of the STCMA for unidirectional frequency conversion. By loading the STC film to the STCMA, an equivalent sinusoidal moving envelope is formed along the length of the waveguide. The STCMA can transform the in-plane guided waves with frequency \({\omega }_{0}\) into the out-of-plane propagating waves with another frequency \({\omega }_{0}-{\Omega }_{E}\) in free space for a forward propagating envelope. b Conversion efficiency of the unidirectional frequency conversion as a function of the frame rate L of the STC film. c–e Comparison among the static envelope (c), forward (d), and backward (e) moving envelopes. There is no frequency conversion for a static envelope, as shown in the dispersion diagram in (c). A forward or backward propagating envelope provides additional momentum along the x-axis and changes its frequency. Due to the phase match to free space, only the downward and upward transitions are allowed for the forward and backward propagating envelopes, respectively. The input and modulated angular frequencies are \({\omega }_{0}=2\pi \times 22.5\) GHz and \({\Omega }_{E}=2\pi \times 10\) kHz, respectively. WG: waveguide; STC: space-time-coding. Enve.: envelope.

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In other words, the amplitude envelope also moves with a velocity of \({v}_{E}\). In contrast, \({v}_{E}=0\) (or \({\Omega }_{E}=0\)) denotes that the moving envelope degenerates into a static one, corresponding to a purely space-only modulated static envelope (Fig. 3c)61,62.

With a sinusoidal moving envelope, the space-time modulated guided wave has the form of:

$${E}_{{gw}}(x,t)= {A}_{E}[1+{M}_{E} \cos \left({\Omega }_{E}t-{k}_{E}x\right)]{e}^{-j{\beta }_{{gw}}x}{e}^{j{\omega }_{0}t} \\= {A}_{E}{e}^{-j{\beta }_{{gw}}x}{e}^{j{\omega }_{0}t}+{A}_{E} \frac{{M}_{E}}{2}{e}^{-j({\beta }_{{gw}}+{k}_{E})x}{e}^{j({\omega }_{0}+{\Omega }_{E})t} \\ +\,{A}_{0} \frac{{M}_{E}}{2}{e}^{-j({\beta }_{{gw}}-{k}_{E})x}{e}^{j({\omega }_{0}-{\Omega }_{E})t}$$
(3)

where \({\beta }_{{gw}}\) is the guided-wave wavenumber. The second and third terms on the right-hand side of Eq. (3) show that the moving envelope results in nonlinear effects. New frequencies \({\omega }_{0}\pm {\Omega }_{E}\) are generated, where ‘\(+\)’ and ‘\(-\)’ denote the upward (or \(+1\)st harmonic order) and downward (or \(-1\)st harmonic order) frequency translations, respectively. The momentums of the newly generated harmonic frequencies are \({\beta }_{{gw}}\pm {k}_{E}\), indicating that the moving envelope results in an additional linear momentum \(\pm {k}_{E}\).

Here, we show that the moving envelope can achieve unidirectional frequency conversion for metasurface antennas. For frequency down conversion to \({\omega }_{0}-{\Omega }_{E}\), the moving envelope should propagate along the +x forward direction with \({v}_{E} \, > \, 0\) (see Fig. 3d). Meanwhile, the wavenumber of the moving envelope \({k}_{E}\) fulfills \(-{k}_{{\omega }_{0}-{\Omega }_{E}} < \, {\beta }_{{gw}}-{k}_{E} \, < \, {k}_{{\omega }_{0}-{\Omega }_{E}}\), where \({k}_{{\omega }_{0}-{\Omega }_{E}}\) is the free-space wavenumber at the target down-convert frequency \({\omega }_{0}-{\Omega }_{E}\). As shown in the dispersion diagram in Fig. 3d, the lower harmonic frequency falls into the light cone and radiates into space, while the higher harmonic \({\omega }_{0}+{\Omega }_{E}\) is not supported in both the free space and waveguide due to the large momentum mismatch. Likewise, when the envelope propagates with \({v}_{E} < 0\) (along the -x backward direction), the spatiotemporal modulation results in unidirectional up-conversion in free space as long as the wavenumber \({k}_{E}\) satisfies \(-{k}_{{\omega }_{0} \! \!+\! {\Omega }_{E}} < \, {\beta }_{{gw}}+{k}_{E} \, < \, {k}_{{\omega }_{0} \! \!+\! {\Omega }_{E}}\), where \({k}_{{\omega }_{0} \! \!+\! {\Omega }_{E}}\) is the free-space wavenumber at the frequency \({\omega }_{0}+{\Omega }_{E}\). Similarly, the downward conversion to frequency \({\omega }_{0}-{\Omega }_{E}\) is forbidden in both free space and waveguide without phase matching, as shown in Fig. 3e. For a conventional static envelope with \({v}_{E}=0\) (Fig. 3c), there is only guided-wave to free-space propagating wave translation without any frequency conversion. In a physical sense, the moving envelope results in an equivalent artificial Doppler shift to the free-space radiation waves compared to conventional static or reconfigurable envelopes.

The implementation of continuous waveform modulation typically relies on continuously tunable active components, such as varactor diodes47,54,56,63,64,65,66,67,68, which provide complete control but are associated with high insertion loss and low modulation speed. Here, we demonstrate that the moving envelope can be achieved by the simplest 1-bit ON-OFF modulation for STCMAs in practice by leveraging its unique waveguide integration property. In analogy to a video comprising multiple still frames appearing as one continuous video, the envelope seems to propagate in a fluid motion as long as the still envelope frames are realized at a fast rate. Therefore, we divide the continuous envelope propagation into several frames with still envelopes, as shown in Fig. 4a and Supplementary Movie 1. The illusion of motion in projected films results in higher-order harmonics generation. The effects of the frame rate (number of frames in one modulation time cycle \({T}_{E}=\frac{2\pi }{{\Omega }_{E}}\)) are investigated in Methods. The conversion efficiency, defined as the power in the target frequency over the total radiated power in all frequencies, versus the frame rate is shown in Fig. 3b. We can observe that the conversion efficiency increases with the increase of the frame rate and is saturated as the frame rate reaches 8 frames per cycle (fpc). In this proof-of-concept example, the modulation frequency is \({\Omega }_{E}=2\pi \times 10\) kHz, and the frame rate is set as 8 fpc to ensure a high-frequency conversion efficiency of 92%. The STCMA will repeat loading these 8 frames after one modulation cycle \({T}_{E}\). For each still envelope frame, we leverage the time-average effect of the space-time modulation at the fundamental frequency to form the equivalent sinusoidal amplitude distribution. This is achieved by loading the corresponding 1-bit digital ‘0/1’ space-time-coding (STC) matrix (see Supplementary Note S5) to the STCMA. Stringing these ‘0/1’ STC matrixes together leads to the ‘STC film’ – a terminology borrowed from the cinematography, as shown in Fig. 4a. Therefore, by continuously loading the STC film to the STCMA, an equivalent moving envelope can be formed along the metasurface antenna aperture.

Fig. 4: Implementation of unidirectional frequency conversion for STCMA.
figure 4

a Synthesis process of the STC film for unidirectional frequency downward conversion. The moving envelope is divided into several still envelope frames; each still envelope is realized by one specific STC matrix. The time-varying STC matrix forms the STC film. b, c The measured radiation patterns (b) and spectrum distribution in the main-beam direction (c) for a forward propagating envelope. d Synthesis process of the STC film for unidirectional frequency upward conversion. e, f The measured radiation patterns (e) and spectrum distribution in the main-beam direction (f) for a backward propagating envelope. The input and modulated angular frequencies are \({\omega }_{0}=2\pi \times 22.5\,\)GHz and \({\Omega }_{E}=2\pi \times 10\) kHz, respectively.

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We utilize the metasurface antenna in Fig. 1b to validate the STC film for frequency transitions. The frequency conversion and radiation performance of the STCMA are characterized in a microwave anechoic chamber (See Methods and Supplementary Fig. S14). Firstly, we demonstrate the unidirectional frequency down-conversion by forming a forward propagating envelope along the waveguide. Figure 4a–c present the 1-bit digital ‘0/1’ STC film, the corresponding measured radiation patterns at different harmonic frequencies, and the measured spectrum distribution in the output direction, respectively. We observe that the STCMA forms a high-directivity beam in the broadside direction (\(\theta={0}^{{{\rm{o}}}}\)) at the target ‒1st harmonic frequency (\({\omega }_{0}-{\Omega }_{E}\)), while the powers of the other undesired harmonics are at least ‒25 dB below that of the target down-converted frequency. Likewise, Fig. 4d–f present the frequency up-conversion scenario by forming a backward propagating envelope. It is evident from Fig. 4e, f that the +1st harmonic frequency (\({\omega }_{0}+{\Omega }_{E}\)) dominates the output free-space waves. These results demonstrate that the STCMA can achieve unidirectional frequency conversion by forming an equivalent moving envelope along the length of the waveguide.

Frequency comb synthesis

In the previous section, we demonstrated that a basic sinusoidal moving envelope corresponds to single-frequency unidirectional conversion and high-directivity radiation for STCMAs. Furthermore, the linear momentum of the generated free-space wave is the sum of the momentums of the guided wave and the moving envelope due to the conservation of momentum. A one-to-one mapping relationship exists between the sinusoidal moving envelope and the converted harmonic frequency in free space. Here, we show that the metasurface antenna can achieve simultaneous multi-frequency conversions or frequency comb generation in free space through the superposition of multiple basic sinusoidal moving envelopes. The total synthetic moving envelope can be expressed as:

$$A(x,t)={A}_{E}\left[1+{M}_{E}{\sum }_{i=1}^{I}\cos \left({\Omega }_{{Ei}}t-{k}_{E}x\right)\right]$$
(4)

where \({\Omega }_{{Ei}}\) is the temporal angular frequency of the ith basic sinusoidal moving envelope, and I is the total number of the basic moving envelopes or the number of the target converted harmonic orders. We first demonstrate the STCMA for dual-frequency transitions (Fig. 5a–c). In this case, the two basic sinusoidal moving envelopes share the same amplitude and temporal frequency but have opposite moving directions with \({v}_{E1}={-v}_{E2}\). The synthetic moving envelope is a type of standing wave, which is the combination of two identical sinusoidal envelopes propagating in opposite directions, as shown in Fig. 5a. We then divide this time-varying standing-wave envelope into multiple still envelope frames, with each still frame realized through loading the corresponding 1-bit ‘0/1’ STC matrix to the metasurface antenna. The synthesized STC film for dual-frequency transition is presented in Supplementary Fig. S7, and the corresponding measured radiation patterns and spectra are given in Fig. 5b, c, respectively. From Fig. 5c, we observe that two target converted harmonic frequencies, \({\omega }_{0}\pm {\Omega }_{E}\) dominate the output spectra, while other undesired higher-order harmonics are highly suppressed in free space. Furthermore, the STCMA launches two almost identical radiation patterns at the frequencies \({\omega }_{0}\pm {\Omega }_{E}\) (Fig. 5b), indicating the perfect decoupling between the two harmonic frequencies.

Fig. 5: STCMA for arbitrary frequency comb synthesis.
figure 5

a The synthetic standing-wave moving envelope for dual-frequency transition. The corresponding STC film is given in Supplementary Fig. S7. b, c The measured radiation patterns at the target harmonic frequencies (b) and the spectrum distribution in the output direction (c) for converting the frequency to \(\omega={\omega }_{0}\pm {\Omega }_{E}\) in free space. d The synthetic moving envelope for quad-frequency transition. The corresponding STC film is given in Supplementary Fig. S8. e, f, The measured radiation patterns at the target harmonic frequencies (e) and spectrum distribution in the output direction (f) for converting the frequency to \(\omega={\omega }_{0}\pm {\Omega }_{E},\,{\omega }_{0}\pm 2{\Omega }_{E}\). g The synthetic moving envelope for arbitrary penta-frequency transition. The corresponding STC film is given in Supplementary Fig. S9. h–i The measured radiation patterns at the target harmonic frequencies (h) and spectrum distribution in the output direction (i) for converting the frequency to \(\omega={\omega }_{0}-4{\Omega }_{E},\, {\omega }_{0}-2{\Omega }_{E},\, {\omega }_{0}-{\Omega }_{E},\, {\omega }_{0}+{\Omega }_{E},\, {\omega }_{0}+3{\Omega }_{E}\). The input and modulated angular frequencies are \({\omega }_{0}=2\pi \times 22.5\,\)GHz and \({\Omega }_{E}=2\pi \times 10\) kHz, respectively.

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Following the same strategy, we are able to achieve quad-frequency translations based on Eq. (4) (Fig. 5d–f). The temporal frequencies of the constituted basic moving envelopes are \({\Omega }_{{Ei}}=\pm {\Omega }_{E},\,\pm 2{\Omega }_{E}\). The total synthetic moving envelope is the superposition of the two standing-wave envelopes, as shown in Fig. 5d. The corresponding measured radiation patterns and spectra are shown in Fig. 5e-f, respectively. We observe that the four target harmonic frequencies are successfully generated in free space with high-directivity patterns. In fact, the converted frequencies are not necessary to be symmetric with respect to the input frequency \({\omega }_{0}\), and frequency comb with arbitrary harmonic orders can be generated by the STCMA. Without loss of generality, the STC film was synthesized to realize penta-frequency transitions, with the temporal frequencies of the constituted basic envelopes being arbitrarily chosen as \({\Omega }_{{Ei}}=- \! 4{\Omega }_{E},\,- \! \! 2{\Omega }_{E},- \! {\Omega }_{E},\, {\Omega }_{E},\, 3{\Omega }_{E}\). Figure 5g–i present the synthetic moving envelope and the corresponding measured radiation patterns and spectra in free space, respectively. Again, the STCMA generates five target frequencies with pencil-beam-like radiation patterns. Five spectral lines dominate the output power of the free-space waves.

Arbitrary harmonic order synthesis

In the process of frequency comb synthesis described above, the output beams with different converted frequencies radiate towards the same direction with identical amplitude to form frequency combs. This requires all the basic sinusoidal moving envelopes to exhibit the same weight ME and spatial frequency \({k}_{E}\). In this section, we release these restrictions such that the STCMA can not just generate multiple harmonic orders but also independently manipulate their other wave properties, including amplitude, momentum, and phase (Fig. 6a). To this end, the synthetic moving envelope in Eq. (4) is extended to

$$A(x,t)={A}_{E}\left[1+{\sum }_{i=1}^{I}{M}_{{Ei}}\cos \left({\Omega }_{{Ei}}t-{k}_{{Ei}}x+{\varphi }_{{Ei}}\right)\right]$$
(5)

where \({M}_{{Ei}}\), \({k}_{{Ei}}\), \({\varphi }_{{Ei}}\) represent the weight (modulation depth), spatial frequency, and initial phase of the ith basic sinusoidal moving envelope, respectively. For each basic sinusoidal moving envelope, the modulation depth, temporal frequency, spatial frequency, and initial phase (\({M}_{{Ei}}\), \({\Omega }_{{Ei}}\), \({k}_{{Ei}}\), \({\varphi }_{{Ei}}\)) determines the amplitude, frequency, momentum, and phase of the radiated harmonic in free space, respectively (See Fig. 6a). Figure 6b illustrates the synthesis process for dual-harmonic control, where two basic sinusoidal moving envelopes are superposed to achieve dual-harmonic radiation and independent manipulation of their wave properties. The synthetic moving envelope can be practically implemented by loading the corresponding STC film to the metasurface antenna as demonstrated in the ‘Single frequency conversion’ section. The same synthesis process can equally be applied to arbitrary harmonic order control for the STCMA.

Fig. 6: STCMA for arbitrary harmonic frequency order conversion and manipulation.
figure 6

a The mapping relationship between the parameters of the moving envelope along the metasurface (\({M}_{E}\), \({\Omega }_{E}\), \({k}_{E}\), \({\varphi }_{E}\)) and the properties of the radiated wave in free space. b The synthetic moving envelope for arbitrary harmonic order independent control by superposing the basic sinusoidal moving envelopes with independent weight, temporal and spatial frequencies, and initial phase. c Measured radiation patterns at the frequencies \(\omega =({\omega }_{0}-9{\Omega }_{E},\,{\omega }_{0}-2{\Omega }_{E},\,{\omega }_{0}+5{\Omega }_{E},\,{\omega }_{0}+12{\Omega }_{E})\), when the beams of the four harmonics scan to −30o with the same amplitude. d Measured radiation patterns at the four harmonic frequencies when the beams scan to −30o with different amplitudes. e Measured radiation patterns at the four frequencies when the beams scan to different directions (−15o, −5o, 5o, +15o) with different amplitudes. f Measured radiation phases at the four harmonics versus the initial phase \({\varphi }_{E12}\) of the basic sinusoidal moving envelope. The measured radiation phases at the four harmonics versus other initial phases \({\varphi }_{E5},\,{\varphi }_{E-2},\) and \({\varphi }_{E-9}\) are shown in Supplementary Fig. S11. In each case, only one initial phase is changed, while other initial phases are kept as 0. g Measured radiation patterns of the 18 harmonics covering −40o to 40o. h Measured radiation patterns of the 18 harmonics covering 0o to 45o. The input and modulated angular frequencies are \({\omega }_{0}=2\pi \times 22.5\,\)GHz and \({\Omega }_{E}=2\pi \times 10\) kHz, respectively.

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In the first illustrative example, the moving envelope is synthesized to perform independent and simultaneous control of four randomly selected harmonic orders with \({\Omega }_{{Ei}}=- \! 9{\Omega }_{E},\,- \! 2{\Omega }_{E},\,+ \! 5{\Omega }_{E},\,+ \! 12{\Omega }_{E}\). Figure 6c shows the measured radiation patterns of the STCMA when the four target harmonic frequencies share the same amplitude at the output angle of \(-\)30o. The measured spectrum at the main beam direction is given in Supplementary Fig. S10a. This is enabled by superposing four basic sinusoidal moving envelopes with different temporal frequencies \({\Omega }_{{Ei}}\), yet an identical weight \({M}_{E}\) and spatial frequency \({k}_{E}\). We then vary the weight ratios among the four basic sinusoidal moving envelopes \({M}_{{Ei}}=\) (0.353, 0.5, 0.707, 1) to control the power distribution of the four converted harmonics with (0.125, 0.25, 0.5, 1), whose measured radiation patterns and spectrum are plotted in Fig. 6d and Supplementary Fig. S10b, respectively. Furthermore, we vary the spatial frequencies of the four sinusoidal moving envelopes \({k}_{{Ei}}\) to steer the output directions of the converted harmonics into different directions (\(-\)15o, \(-\)5o, +5o, +15o), as shown in Fig. 6e. We further tune the initial phase of the constitutive sinusoidal moving envelope to change the radiation phase of the harmonics. Figure 6f shows the radiated phases of the four harmonics as a function of the initial phase \({\varphi }_{E12}\). The radiated phases versus the other initial phases (\({\varphi }_{E-9},\, {\varphi }_{E-2},\, {\varphi }_{E5}\)) of the sinusoidal moving envelopes are given in Supplementary Fig. S11. In each case, only one initial phase is changed, while other initial phases are kept as 0. It can be observed that the radiation phase at one harmonic frequency is merely changed by the corresponding initial phase of the sinusoidal moving envelope. Therefore, the radiation phase of each harmonic can be independently tuned by the corresponding initial phase of the fundamental sinusoidal envelope.

To demonstrate the flexibility of our STCMA for arbitrary harmonic synthesis, we synthesize a moving envelope based on Eq. (5) to achieve simultaneous transitions and independent controls of 18 harmonic frequencies. The measured radiation patterns of the 18 harmonics \({\omega }_{0}\pm i{\Omega }_{E}\), \(i=1,\cdots,\,9\) covering main beam directions from \(-\)40o to +40o are presented in Fig. 6g. To show the independent harmonic control, the 18 harmonic beams are further steered to cover the forward angular region from 0o to +45o (Fig. 6h). Note that the synthetic metasurface antenna can generate arbitrary harmonic orders instead of arbitrary high frequencies (see the difference between high-order harmonic and high frequency in Supplementary Note S4). Limited by the finite switching speed of the PIN diodes and the FPGA control speed, the maximum conversion frequency is around \({\omega }_{\max }={\omega }_{0}+2\pi \times 5\) MHz for the metasurface antenna prototype. Nevertheless, it is important to emphasize that previously reported spatial multiplexing solution59 cannot achieve arbitrary harmonic order generation and independent control, as introduced in Supplementary Note S3.

Frequency division multiplexing

Due to the arbitrary harmonic order controllability, we show the application of the synthetic metasurface antenna for FDM communications (Fig. 7d), i.e., simultaneous transmission of multiple signals over a single communication channel leveraging the frequency diversity. We start from the direct generation of single-tone radiation carrying one data stream. Based on the mapping relationship between the moving envelope and its radiation (Fig. 6a), the amplitude and phase of the baseband information \(S\left(t\right)\) should be encoded into the weight and initial phase of the moving envelope, respectively. Therefore, the moving envelope in Eq. (1) is modified as follows to carry the baseband information S(t)

$${A}_{s}(x,t)={A}_{E}[1+{M}_{E}\left|S\left(t\right)\right|\cos ({\Omega }_{E}t-{k}_{E}x+\angle S\left(t\right))]$$
(6)
Fig. 7: STCMA for direct data transmission in FDM applications.
figure 7

a Block diagram of the STCMA to directly generate 16 QAM modulated waveform. The baseband symbols map to the rolls of STC film, controlling the STCMA to radiate an information-carrying beam in free space. b Measured decoded constellation diagram of the 16 QAM signal at the receiver end at frequency \({\omega }_{0}+{\Omega }_{E}\). c Experimental setup of the wireless communication testbed based on the STCMA. d Conceptual illustration of the STCMA for FDM applications. By continuously loading STC films, the STCMA can directly generate multiple independent signals at different harmonics pointing to users located at different positions. e Measured decoded constellation diagrams and Q-eye diagrams for the receivers located at \(\theta=({-30}^{{{\rm{o}}}},\, {0}^{{{\rm{o}}}},\, {30}^{{{\rm{o}}}})\) at frequencies \(({\omega }_{0}+{\Omega }_{E},{\omega }_{0}+{2\Omega }_{E},\, {\omega }_{0}+{3\Omega }_{E})\), respectively. More measured results uisng different modulation schemes are given in Supplementary Fig. S12.

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In this manner, the moving envelope can directly generate information-carrying radiation in free space. Here, we take the direct generation of 16 quadrature amplitude modulation (QAM) radiation as an illustrative example (see Fig. 7a). We first synthesize 16 different basic sinusoidal moving envelopes corresponding to the 16 baseband symbols with different combinations of the baseband amplitude and phase information using Eq. (6). These 16 basic sinusoidal moving envelopes can be implemented by the corresponding 16 rolls of STC films loading to the STCMA. Therefore, the information-carrying modulated waveform in free space can be generated by loading the time-dependent STC films to the STCMA, i.e., switching to the corresponding STC film roll whenever a different symbol is sent. For further demonstration, we build a wireless communication testbed and conduct the over-the-air measurement (See Fig. 7c, Methods, and Supplementary Movie 2). In the communication system, our synthetic metasurface antenna serves as the transmitter to directly radiate the modulated waveform with a monochromatic wave input. The demodulated constellation diagram at the receiver end is presented in Fig. 7b with an error vector magnitude (EVM) of 6.83% and a transmission data rate of 40 kbps.

To multiplex I harmonic frequencies carrying different unrelated information stream \({S}_{i}\left(t\right)\), the moving envelope should be

$${A}_{s}(x,t)={A}_{E}\left[1+{\sum }_{i=1}^{I}{M}_{{Ei}}\left|{S}_{i}\left(t\right)\right|\cos \left({\Omega }_{{Ei}}t-{k}_{{Ei}}x+\angle {S}_{i}\left(t\right)\right)\right]$$
(7)

The synthetic moving envelope is the superposition of the I basic sinusoidal moving envelope; each basic sinusoidal moving envelope carries one baseband stream. Figure 7d shows the conceptual illustration of the synthetic STCMA for FDM wireless communications. By continuously loading the STC film pertinent to the baseband message, the STCMA can simultaneously generate multiple information-carrying beams at different harmonic frequencies pointing to users located at different positions.

To illustrate the STCMA for direct data transmission with FDM, we built a tri-channel wireless communication system in which the STCMA generates three beams at frequencies \({\omega }_{{Ei}}=({\omega }_{0}+{\Omega }_{E},\, {\omega }_{0}+{2\Omega }_{E},\, {\omega }_{0}+{3\Omega }_{E})\) pointing to different directions \({\theta }_{i}=({-30}^{{{\rm{o}}}},\, {0}^{{{\rm{o}}}},\, {30}^{{{\rm{o}}}})\). We apply different modulation schemes (BPSK, QPSK, and 8PSK) to the three harmonics to show the capability of the synthetic STCMA in carrying unrelated baseband information. Figure 7e presents the measured decoded constellation diagrams and eye diagrams for the receivers located at different locations with the corresponding frequency. Here, the EVM is considered as the figure of merit for direct data transmission quality. The measured EVMs for the three channels are 6.49%, 6.51%, and 7.28%, with the transmission data rate of 10 kbps, 20 kbps, and 30 kbps, respectively. More measured results for other modulation schemes are given in Supplementary Fig. S12. It is important to point out that due to the arbitrary harmonic synthesis flexibility of the STCMA, the number of the generated beams (or harmonic frequencies) I can be easily added or dropped in a software control manner subject to the number of users in the practical scenario. With a monochromatic wave input, the synthetic STCMA can directly generate FDM signals in free space, avoiding the requirement of complicated RF chains in conventional superheterodyne-based FDM transmitters, including mixers, digital-to-analog converters (DACs), filters, phase shifters, power dividers, and multiplexers.

Discussion

In summary, we have proposed a harmonic synthesis solution allowing STCMAs to generate and independently manipulate arbitrary harmonic orders simultaneously. The synthesis of the moving envelope involves superposing multiple basic sinusoidal moving envelopes, each corresponding to one specific target harmonic frequency conversion and wave properties manipulation. To realize the synthetic moving envelope, we have adapted the ‘STC film’ concept from the video production, which divides the moving one into various still frames; each frame is realized by loading the corresponding STC matrix to the metasurface antenna. As proof-of-concept examples, we demonstrated our synthetic STCMA for the realization of single-frequency unidirectional transition, arbitrary frequency comb generation, arbitrary harmonics independent manipulation, as well as their application in FDM wireless communications. Intriguingly, all these complicated harmonics controls are realized by the 1-bit ON-OFF switching of meta-atoms between the radiating and non-radiating states. Furthermore, all other undesired harmonics are highly suppressed in free space due to the sideband-free nature of the waveguide-integrated STCMA.

Our developed synthetic moving envelope approach and its implementation substantially improve the controllability of metasurfaces and metamaterials in the spectral and spatial domains. The proposed concept and methodology can be extended to terahertz and optical frequencies by leveraging alternative active control materials. The unprecedented frequency controllability of our synthetic STCMA, together with its salient features of sideband-free, simple coding strategy (1-bit), and seamless integration with source, make it highly promising for various applications, such as next-generation wireless communications, spectroscopy, metrology, arbitrary waveform generation, and quantum information science. With the potential for significant impact across multiple fields, we anticipate that our proposed approach will trigger further innovation in the design and application of metamaterials and metasurfaces.

Methods

Effects of the frame rate

The continuously propagating envelope consists of a series of still frames. The realization of these still envelope frames at a certain speed (frame rate) in proper order leads to the appearance of motion. Different frame rates will yield different frequency conversion results for metasurface antennas. Suppose that the moving envelope consists of \(L\) still frames in one modulation cycle \({T}_{E}\), corresponding to the frame rate of L fpc. The moving envelope in one modulation period \({T}_{E}\) can be written as a linear combination of L still frames:

$$A(x,t)={A}_{E} \left \{1+{M}_{E}{\sum }_{q=1}^{L}\cos \left[q\frac{2\pi }{L}{U}_{q}\left(t\right)-{k}_{E}x\right]\right\}$$
(8)

where \({U}_{q}\left(t\right)\) is the qth pulse function in one modulation period \({T}_{E}\), given as

$${U}_{q}\left(t\right)=\left\{\begin{array}{c}1,\,(q-1)\frac{{T}_{E}}{L}\le t\le q\frac{{T}_{E}}{L}\\ 0,\hfill {else}\,\end{array}\right.$$
(9)

The pulse width of \({U}_{q}\left(t\right)\) is \(\frac{{T}_{E}}{L}\), indicating the duration or length of one still frame. Since \(A(x,t)\) is a periodic function of time, we decompose it into Fourier series:

$$A\left(x,t\right)={\sum }_{m=-{{\infty }}}^{{{\infty }}}{a}_{m}{e}^{{\, \, jm}{\Omega }_{E}t}$$
(10)

The Fourier coefficient \({a}_{m}\) at the mth harmonic frequency is given by

$${a}_{m}(x)=\frac{1}{{T}_{E}}{\int }_{0}^{{T}_{E}}A(x,t){e}^{-{jm}{\Omega }_{E}t}{dt}$$
(11)

Substituting Eq. (8) into (11), we obtain (see Supplementary Note 1 for the detailed derivations):

$${a}_{m}\left(x\right)={A}_{E}\delta \left(m\right)+{A}_{E}\frac{{M}_{E}}{L}{\sum }_{q=1}^{L}\cos \left(q\frac{2\pi }{L}-{k}_{E}x\right){sinc}\left(\frac{\pi m}{L}\right){e}^{-j\pi m(2q-1)/L}$$
(12)

The spatiotemporal modulated guided wave with an envelope frame rate of L fpc can be written as

$${E}_{{gw}}(x,t)={A}_{E}{\sum }_{m=-{{\infty }}}^{{{\infty }}}\left[\delta \left(m\right)+\frac{{M}_{E}}{L}{\sum }_{q=1}^{L}\cos \left(q\frac{2\pi }{L}-{k}_{E}x\right){sinc}\left(\frac{\pi m}{L}\right){e}^{-j\pi m(2q-1)/L}\right]$$
$$\cdot \, {e}^{-j{\beta }_{{gw}}x}{e}^{j({\omega }_{0}+m{\Omega }_{E})t}$$
(13)

From Eq. (13), we observe that a finite frame rate results in the generation of higher-order harmonics in addition to the \(m=0,\pm 1\) harmonics for a continuously propagating moving envelope. Note that not all these higher-order harmonics will radiate into free space due to the phase mismatch. The radiation pattern of the metasurface antenna at the mth-order harmonic frequency can be obtained by taking the spatial Fourier transfer of Eq. (13):

$${E}_{m}\left(\theta \right)={{\mathscr{F}}} \left\{\left[\delta \left(m\right)+\frac{{M}_{E}}{L}{\sum }_{q=1}^{L}\cos \left(q\frac{2\pi }{L}-{k}_{E}x\right){sinc}\left(\frac{\pi m}{L}\right){e}^{-\frac{j\pi m\left(2q-1\right)}{L}}\right] {A}_{E}{\Pi (x)e}^{-j{\beta }_{{gw}}x} \right\}$$
(14)

where \(\Pi (x)\) is the rectangular function corresponding to the finite length of the metasurface antenna. For the unidirectional down conversion, the frequency transition efficiency of the metasurface antenna can be calculated by

$$\eta=\frac{{\int }_{0}^{2\pi }{\int }_{0}^{\pi }{{{\rm{|}}}{E}_{m=-1}(\theta ){{\rm{|}}}}^{2}\sin \theta d\theta d\varphi }{{\sum }_{m=-{{\infty }}}^{{{\infty }}}{\int }_{0}^{2\pi }{\int }_{0}^{\pi }{{{\rm{|}}}{E}_{m}(\theta ){{\rm{|}}}}^{2}\sin \theta d\theta d\varphi }$$
(15)

where the numerator and denominator represent the radiated powers of the target \(m=-1\) harmonic and all harmonics of the metasurface antenna, respectively. Without loss of generality, we consider an STCMA with a length of 6.15\({\lambda }_{0}\), where \({\lambda }_{0}\) is the free-space wavelength at the input frequency. The moving envelope is synthesized to achieve unidirectional frequency down-conversion and radiate at the broadside direction (\(\theta={0}^{{{\rm{o}}}}\)). To investigate the effects of frame rate for frequency translations, Supplementary Fig. S13 shows the radiation patterns of the metasurface antenna calculated by Eq. (14) when the adopted frame rates are 2 fpc, 4 fpc, 8 fpc, and 16 fpc. We can observe that the power level of the undesired harmonics (sideband) reduces with the increase in frame rate. Based on the radiation patterns at different harmonics, we can obtain the frequency conversion efficiency as a function of the frame rate based on Eq. (15), and the results are given in Fig. 3b. It is evident that a higher frame rate results in higher conversion efficiency. Specifically, the conversion efficiency reaches 92% as the envelope frame rate is 8 fpc.

Experimental verification

To validate the synthetic moving envelope for arbitrary harmonic manipulation, we implement an STCMA consisting of 41 elements. The configuration of the meta-atom is presented in Fig. 1c with detailed geometrical parameters. A low-cost FPGA (ALTERA Cyclone IV) control board is adopted to provide 41 independent control signals to control the radiating states of the meta-atoms. The two ±45o-oriented slots in each meta-atom are in the same radiating states such that the polarization of the radiated waves is linear x-polarization. The switching speed of the PIN diodes is 10 MHz, and the modulation angular frequency is set as \({\Omega }_{E}=2\pi \times 10\) kHz in this proof-of-concept design.

A metasurface antenna was fabricated based on the commercial printed circuit broad (PCB) technology, as shown in Fig. 1b. The radiation performance of the STCMA was measured in a microwave anechoic chamber (see Supplementary Fig. S14). A microwave signal generator (Agilent E8267D) is adopted as the excitation source to provide the monochromatic signal \({\omega }_{0}=2\pi \times 22.5\,\)GHz to feed the metasurface antenna. A linearly polarized diagonal horn connected to a vector network analyzer (VNA, Keysight N9041B) is adopted as the receiver to detect the waves radiated from the metasurface antenna. The diagonal horn is mounted onto a reconfigurable robotic arm, which can flexibly rotate along a user-defined circular trajectory to measure the radiation pattern of the metasurface antenna.

For the communication testbed (see Fig. 7c), a microwave signal generator is used to launch a continuous wave with a frequency of \({\omega }_{0}=2\pi \times 22.5\,\)GHz to feed the STCMA. Random binary bit streams with 16 QAM modulation scheme are mapped to the corresponding STC films. By continuously loading the STC film to the FPGA, the synthetic STCMA can directly launch the information-carrying waveform in free space. A linearly polarized diagonal horn is utilized as the receiving antenna and connected to a vector signal analyzer (VSA, Keysight N9041B). The VSA can demodulate the RF signals and provide the key performance indicators, including real-time constellation diagram, eye diagram, EVM, and signal-to-noise ratio (SNR). The distance between the STCMA and receiving horn antenna is 1.4 m, and the input power of the STCMA is \(-\)10 dBm. In the FDM communication testbed, the receiving horn was manually moved to the main beam direction of different harmonic frequencies since only one VSA is available in the lab.