Utilization of harmonics in phaseless near-field microwave computational imaging based on space–time-coding transmissive metasurface

Imaging objects through microwaves possesses nonionization and penetrability capabilities compared with infrared, visible light, and x-ray. Therefore, microwave imaging technology was comprehensively investigated for security inspection, e.g., concealed weapon detection and whole-body screening. For such a near-field scenario, microwave holographic imaging was proposed based on the range migration technique.^1–5 Furthermore, a general holographic inversion method accelerated by range stacking for linear frequency modulation radar was presented in Refs. 6 and 7. Other methods inspired by the optical domain, such as cubic phase element,⁸ tomography,⁹ and incoherent imaging,¹⁰ were also implemented in the microwave domain. When we separate the imaging procedure as data acquisition and data post-process, the computational imaging (CI) framework is established, bridging signal processing and microwave imaging.¹¹ Under this viewpoint, existing imaging methods can be described in a compact mathematical form. Then, the compressive sensing (CS) technique is employed to solve the underdetermined problem and to improve the imaging speed and quality.^12,13 Note that the above-mentioned studies usually require the raster scanning technique or complex distribution apertures.^11,14,15

To reduce the synthetic aperture of raster scanning and multiple detectors, single-pixel imaging via CS was first proposed and realized in light and terahertz bands.^16,17 In 2014, by leveraging the frequency-diverse metamaterial aperture, a single-pixel microwave imager was developed and experimentally validated.¹⁸ Moreover, various metasurfaces, which are two-dimensional metamaterials, were utilized to realize the CI systems.^19–21 Among these metasurfaces, dynamic and programmable metasurfaces gain much attention due to the reconfigurability of sensing modes.^22–24 By introducing tunable components, e.g., PIN diode, varactor, and microelectromechanical system (MEMS), one could achieve substantial sampling patterns on the inspected plane for coherent or incoherent imaging.^25,26 We mention that the published research works on metasurface-based CI systems mainly operate in the frequency domain without exploiting the time-periodic signals. Recently, space–time-coding (STC) metasurfaces²⁷ that periodically manipulate the digital states of each meta-atom were applied in new architecture wireless communication,^28–30 frequency-modulated continuous waves radar,³¹ electromagnetic nonreciprocity,^32,33 and spectral camouflage.^34,35 In Ref. 36, the time-modulated technique was adopted to characterize materials assisted by metasurface. Hitherto, the near-field sensing modes using harmonics enabled by the STC metasurface are not investigated in a microwave CI system.

Here, based on a transmissive metasurface (TMS) with a proposed STC scheme, we utilize the harmonics in a phaseless microwave CI system. The meta-atom of the TMS realizes a 1-bit amplitude-reconfigurable operation using the digital idea.³⁷ By applying the proposed STC scheme, which periodically flips the 12 coding blocks of a space coding, the TMS produces randomly distributed near-field harmonics that satisfy the CS theory. Combined with two standard horn antennas and the TMS, we build a CI system with single-frequency excitation and a single-pixel detector. The plane wave angular spectrum (PWAS) method that had been validated in our previous work^24,38 is extended to calculate the near-field sensing harmonics. A TMS prototype and the corresponding CI system were experimentally demonstrated. The reconstruction of metallic objects proves that the harmonics generated by the STC scheme are sufficient to achieve a phaseless CI system with low-cost hardware.

As illustrated in Fig. 1, the proposed phaseless near-field CI system consists of a transmitter (Tx), a receiver (Rx), and a TMS. Both the Tx and Rx are standard rectangular horn antennas in K-band. The radiation apertures of Tx, Rx, and TMS are parallel to the xy-plane and located at z = −D′, z = 0, and z = 2D, while the inspected plane locates at z = D. The operation principle of the proposed CI system is described as follows. First, the Tx connected to a signal generator excites the TMS using a monochromatic signal at 21 GHz. Then, after passing across the TMS modulated with a period T, the generated near-field harmonics sample the objects on the inspected plane. The sensing matrix is denoted by H whose row vectors are the sampling harmonics. Next, behind the inspected plane, the Rx connected to a spectrum analyzer measures power levels of harmonics, which is denoted by a column vector of g. We mention that the measured signals are real numbers without phase information. Finally, assuming the object to be imaged is f, in the CI framework, the above procedure can be expressed as

Since g can be readily measured in a single-pixel manner, we should model the sensing matrix for imaging restoration.

Prior to commencing the modeling of the CI system, we present the programmable meta-atom. As shown in Fig. 2(a), the proposed programmable meta-atom has one PIN diode and four layers. The receiving layer and ground (GND) layer are etched on a substrate with a relative permittivity of 3.5 and a thickness of 0.762 mm. The biasing layer and transmitting layer are etched on a substrate with the same parameters. Between the GND layer and the biasing layer, there exists a prepreg substrate with a relative permittivity of 4.2 and a thickness of 0.08 mm. When the PIN diode is turned off, the receiving layer reflects the incident waves, indicating a coding “0” transmission amplitude τ⁰. When the PIN diode is turned on, the incident waves can be conveyed to the transmitting layer for radiation, indicating a coding “1” transmission amplitude τ¹. We note that the presented meta-atom is modified from a phase coding meta-atom, which was used for beam-scanning.^39–42 Structural parameters are annotated in Fig. 2(a). The adopted PIN diode and the simulation method had been detailed in our previous study.²⁴ We report the simulated transmissive magnitude of the τ⁰ and τ¹ coding states in Fig. 2(b). The simulation model is also illustrated as an inset in Fig. 2(b). As can be seen, at 21 GHz, the proposed programmable meta-atom has a 14-dB amplitude modulation depth, meeting the STC need for harmonics operations.

Based on the proposed programmable meta-atom, we elaborate on the modeling of the near-field harmonic distribution of the TMS with the proposed STC scheme. Supposing that the TMS contains M × N programmable meta-atoms with sizes of p_x × p_y along the x-direction and y-direction, respectively; thus, on the inspected plane z = D, the time-domain near-field distribution can be expressed as

where k₀ is the free space wavenumber at 21 GHz, $EyTx$(x_m, y_n, D) stands for the incident tangential electric field from Tx at each element, τ_mn(t) is the STC transmission function, and C(u) and S(u) are standard Fresnel integrals with upper limits, defined as

Equation (2) is extended from the time-independent PWAS method⁴³ that had been applied to analyze an adaptively smart programmable metasurface.³⁸ 

We further discuss the proposed STC scheme. As the STC theory implies,²⁸ we can divide the τ_mn(t) function in Eq. (2) into two dimensions, i.e., space-coding dimension and time-coding dimension. The space-coding dimension is the distribution of τ⁰ and τ¹, while the time-coding dimension is the time-modulated scheme in a period T. Consider a TMS consisting of 12 × 12 programmable meta-atoms presented above, we group and index 12 coding blocks as shown in Fig. 3(a). In the space-coding dimension, each coding block contains one randomly distributed τ¹ and another 11 τ⁰. We illustrate one possible space-coding distribution in Fig. 3(b). In the time-coding dimension, the period T is divided into 12 equal-time slices that correspond to the 12 coding blocks, and each indexed coding block sequentially flips the block's space-coding states (τ⁰ to τ¹ or τ¹ to τ⁰) at each time slice. Under such an STC scheme, the near-field qth (q = 0, ±1, ±2, …) order harmonics on the inspected plane based on Eq. (2) can be determined by

where $τmnq$ is the Fourier series coefficients of τ_mn(t), i.e.,

where I represents the total time slice and $τmni$ constructs the STC matrix whose dimension is M × N × I. We mention that the above STC scheme is described by the STC matrix in Eq. (5), and the Fourier series coefficients for each harmonic are calculated by the STC matrix.²⁸ In Fig. 3(c), we show the STC matrix for the space coding shown in Fig. 3(b). The calculated harmonics including fundamental and qth (q = ±1, ±2, ±3, ±4) harmonics on the D = 150 mm inspected plane are plotted in Fig. 3(d).

From the figures, the significance enabled by the STC technique is twofold. On the one hand, the qth order harmonics present distinct patterns compared with the fundamental harmonic, which can be utilized for CI. On the other hand, the generated harmonics remarkably increase the sensing modes using one randomly distributed space coding compared with the dynamic metasurfaces, which use one randomly distributed space coding as one sensing mode.²² We mention that the peak intensities of the qth (|q| > 0) order harmonics are about −10 dB lower than that of the fundamental harmonic. As the harmonic index increases, the power level will decrease accordingly. In addition, due to the symmetry of the STC scheme, it is recognized that there also exists symmetry of positive order harmonic and the corresponding negative order harmonic, i.e., q = 1 and q = −1. Nevertheless, the presented simple STC scheme suffices to demonstrate the CI system based on harmonics utilization.

A 12 × 12 TMS prototype was fabricated and experimentally examined. As shown in Fig. 4(a), we present the front view (receiving layer) and back view (transmitting layer). The aperture size of TMS is 150 by 100 mm, containing 12 × 12 programmable meta-atoms and 12 sockets for biasing. The 12 biasing sockets representing the 12 coding blocks are connected to an field-programmable gate array (FPGA) control board for independent time modulation of each block. We mention that all the biasing lines are tuned to an identical length for time synchronization. The transmission magnitudes of the fabricated 12 × 12 TMS in all τ⁰ and all τ¹ states were measured and simulated, as shown in Fig. 4(b). The model for both measurement and simulation is also depicted as an inset in Fig. 4(b). The results are obtained from the |S₂₁| of the two horn antennas. Note that before measuring all τ⁰ and all τ¹ states of the fabricated TMS, we measured the |S₂₁| without the TMS for calibration. As can be observed from the results, the fabricated TMS shows good performance in modulating the transmission amplitude, which agrees well with the simulated ones from 20 to 23 GHz. However, there exists disagreement between 19 and 20 GHz, which may be explained by the fabrication errors. Nevertheless, because our CI system only works at 21 GHz, the disagreement between 19 and 20 GHz would not affect the results of imaging reconstruction.

In Fig. 5, we compare the calculated, simulated, and measured near-field distributions on the inspected plane at 21 GHz for four different space coding according to the proposed STC scheme. The distance parameters are D′ = 38 mm and D = 150 mm. The inspected plane is 300 × 300 mm². From the results, it is observed that the calculated normalized patterns are in accordance with the simulated and measured ones, demonstrating the effectiveness of the adopted method. Note that the simulated |E_y| patterns were analyzed by the Ansys Electronics Desktop. The measured |E_y| results were obtained in an anechoic chamber with the same configuration for simulations and calculations. We also mention that existing full-wave simulation tools cannot analyze an STC metasurface, which requires a time-varying boundary condition; therefore, only the space coding results are compared. Nonetheless, the reported comparisons provide the prerequisite for the TMS-based CI system utilizing harmonics.

Having the above description, we show how to build the sensing matrix H using randomly distributed harmonics. We suppose that there exist G × K equal spacing pixels, total L randomly distributed space coding, and total Q harmonics for each space coding. Therefore, each equation in Eq. (1) can be expressed as

where f_gk is the transmissive coefficient of the object to be imaged and h_gk is proportional to $Egkq$·$EgkRx$⁠, where $EgkRx$ is the Rx field at each pixel. Putting the available harmonics together, the complete expression of Eq. (1) is

Since the sensing matrix has been obtained, we further conduct the imaging reconstruction experiments.

In Fig. 6, we show the experimental configuration of the CI system utilizing harmonics generated by the STC TMS. The distances of the parallel planes are D′ = 38 mm and D = 150 mm. The Tx is connected to a signal generator, and the Rx is connected to a spectrum analyzer. The power level of the single-tone signal at 21 GHz was amplified and calibrated to 30 dBm. A thin foam with relative permittivity near unity was applied to support the metallic objects on the inspected plane. The inspected plane is 300 × 300 mm², which is pixelated by 26 × 26, i.e., G = K = 26. Note that a single-pixel spans about 0.8λ₀ × 0.8λ₀, where λ₀ is the wavelength at 21 GHz. Two separated metallic objects spanning 2 × 2 pixels were made by copper foil sheet and glued to the supporting foam. The TMS is modulated by an in-house designed FPGA control board whose modulation time could achieve 1 μs for each output channel independently. In this work, we set the period T = 12 μs; therefore, each coding block flips its coding state in 1 μs duration time while keeping its coding state in the other 11 μs duration time. The CI system is surrounded by microwave-absorbing materials to reduce the multipath reflections.

Considering the total 676 pixels to be imaged, we use L = 50 groups of randomly distributed space coding combined with Q = 9 near-field harmonics, whose harmonic indexes range from 0 to 8. The sparse sampling ratio is about LQ/GK = 0.66. The reflectivity of the target objects to be imaged is plotted in Fig. 7(a). The adopted 50 groups of space coding were generated and programmed to the FPGA, which sequentially flips the coding state of each coding block as the proposed STC scheme. We observed and recorded the power levels of the received spectrum from the spectrum analyzer. The power levels of the 450 groups for image reconstruction are plotted in Fig. 7(b). Power levels of higher harmonics greater (q ⩾ 9) are quite low and, therefore, are dropped. Because we could only measure the phaseless or intensity-only data, we apply the sparse Wirtinger Flow reconstruction algorithm, which has been used to realize phaseless CI in the microwave domain⁴⁴ and optical domain.⁴⁵ Because the unknowns in Eq. (7) are transmissive coefficients, the power levels without the objects were also measured to convert the transmissive results into reflective results, which had been applied in Ref. 46. The reconstructed results are reported in Fig. 7(c). As can be seen, we reconstruct the desired objects with good amplitudes compared with the background at the correct positions.

The proposed TMS can also operate without space–time modulation; thus, frequency domain measurements through a vector network analyzer were conducted using the same 50 groups' space coding. We emphasize that such measurements could obtain complex data, including amplitude and phase information. The reconstructed imaging using the complex data is shown in Fig. 7(d). From the results, the objects can be recognized; however, the quality is poor due to the limited sampling modes. The mean square errors of the results of phaseless signals and complex signals compared with the ideal results are 0.0145 and 0.0453. The comparison proves that by utilizing the near-field harmonics, we can increase the imaging quality with fewer space-coding patterns.

In summary, we have presented the utilization of harmonics in the phaseless near-field CI system based on an STC TMS. The amplitude modulation programmable metasurface is designed in detail. An STC scheme that is simple and effective to launch randomly distributed sensing harmonics is proposed and validated. This study has identified the effectiveness of employing harmonic spectra in a CI system. The insights gained from this study may be of assistance to developing novel imagers in the microwave domain. A further study should focus on a more efficient STC scheme that generates equivalent intensity harmonics for higher mode utilization.

This work was supported by the National Key Research and Development Program of China under Grant No. 2021YFA1401001, the National Natural Science Foundation of China under Grant Nos. 62288101 and 62001342, Key Research and Development Program of Shaanxi under Grant No. 2021TD-07, the Joint Foundation of Key Laboratory of Shanghai Jiao Tong University-Xidian University, Ministry of Education under Grant No. LHJJ/2020-02, and the China Postdoctoral Science Foundation under Grant No. 2021M692496.