Holography is defined as a reconstruction progress for the target object’s image information, which is one of the most promising imaging techniques to record the intensity and phase information [1]. Traditionally, hologram technologies are mostly modulated by electromagnetic waves propagating through the substrate material with a long distance to change the local phase variation for beamforming, appearing to be the defects of low resolution, limited image quality, and bulky profiles comparing to the operation frequency [2]. However, with the appearance of computer-generated holography (CGH) [3], the revolutionary holographic information technology of three-dimensional object patterns based on the diffraction theory can be developed by numerical calculations, attracting feverish research interest owing to applications in display [4], security check [5], multidimensional data storage [6,7], etc.

Metasurfaces [8], different from traditional materials, comprising periodic or aperiodic subwavelength unit artificial electromagnetic (EM) structures, possess the extraordinary physical characteristics to realize the arbitrary wavefront phase control, which are not determined by the inherent properties of their chemical components, but depend on the EM properties of the constituent units. Meanwhile, abrupt phase changes produced by generalized Snell’s law are innovatively developed into the flexible manipulation of amplitude [9–13], phase [14–17], and polarization among EM waves [18]. Evolving from bulky three-dimensional (3D) metamaterials, two-dimensional (2D) metasurfaces with reduced longitudinal composition dimension advantageously featured by ultra-thin profiles, low cost, and low losses, have enormously attracted researchers’ attention on versatile functionalities, for instance, flat lenses [19], EM vortex wave modulators [20–23], thin cloaking [24], radome [25], and metasurface holograms [26,27]. Nevertheless, metasurfaces, according to the different modulation principles, can be divided into spatial wave metasurfaces and surface wave metasurfaces. Conventionally, spatial wave metasurfaces involving reflectarrays [28] and transmitarrays [29], combining the advantages of planar array antennas and parabolic antennas, can be constructed with indispensable air-feeding based on the ray-tracing theory and the gradient phase compensation method, which have been rapidly developed to complete beam scanning, polarization conversion, and wireless energy transmission in the field of communications and radar detection. Namely, the prevalent metasurface hologram concept is defined and implemented based on spatial-wave modulation metasurfaces [26,30–32]. Moreover, considering the modulation freedom in amplitude, phase, polarization, and frequency, metasurface holograms can be categorized into different types on the basis of their ability of information multiplexing, including frequency multiplexing [33,34], polarization multiplexing [35–38], angle multiplexing, direction multiplexing [39], distance multiplexing [40–42], orbital angular momentum (OAM) multiplexing [43–47], along with coding metasurface holograms [48–52]. Apparently, the above-mentioned metasurface holograms are all based on spatial-wave modulation methods with additional feed source, increasing system profile, decreasing integration, and raising assembly error.

Compared to spatial-wave modulated metasurfaces with high transmittance and phase compensation range of 360 deg, the proposed anisotropic impedance holographic metasurfaces (AIHMs) [53] have been inherently demonstrated to consist of meta-atoms with less than one-third of the operation wavelength without considering any reflection or transmission properties, enabling more refined EM regulation. Furthermore, apart from isotropic impedance holographic metasurfaces (IIHMs), AIHMs have higher polarization modulation freedom and polarization isolation owing to their tensor characteristics among the composed meta-atoms, expecting to achieve richer polarization holograms multiplexing function. The traditional anisotropic holographic metasurfaces mainly focus on the radiation regulation of the far-field region, including vortex electromagnetic wave [54], circular polarization wave [55], no diffraction wave [56], and multibeam radiation [57]. The above modulation methods are based on the classic design of circular or square patch with rectangular slot on the diagonal of the anisotropic meta-atom to modulate the cylindrical surface wave. The core is that the angle of the maximum impedance is consistent with the angle of the slotted gap, which greatly reduces the complexity of the design. In this paper, AIHMs depending on the surface-wave modulated principle and leaky-wave theory are innovatively introduced into the holographic imaging category in the near-field region for the first time as known to authors in the existing references at microwave frequencies, as principally described in Fig. 1.

As we know, in conventional optical algorithms the electromagnetic wave propagation formula is usually described based on the Fraunhofer diffraction of the side axis approximation. For holographic surfaces operating in the optical band and containing millions of meta-atoms, this method can greatly reduce the time required for numerical calculations. However, the holographic imaging frequency in this paper is in the microwave band rather than the optical band. Considering factors such as device integration, there are no specific requirements for the duration of numerical simulation. Due to spatial size constraints, the imaging focal length in the microwave band does not yet meet the conditions for paraxial approximation compared to the operating wavelength. Therefore, the formula used in this paper to describe electromagnetic wave propagation is not Fraunhofer diffraction, but rather uses a precise point source propagation function, known as the Green’s function.

The multipoint focusing mechanism for imaging principle on the basis of the surface-wave modulation method has been depicted in Fig. 1. The desired hot spots distributed on the holographic image plane constituted of M×N lattices with grid size d, which totally locates at the X′O′Y′ plane. r⇀mn and R⇀uv respectively refer to the position vectors from the origin O′ to the mnth pixel point and the central position of the uvth meta-atom; r⇀uv is considered as the position vector of the uvth meta-atom relative to the XOY coordinate system. Meanwhile, to realize the desired focal intensity pattern in the near-field region, the field at the resolution of each meta-atom can be calculated by superimposing the fields of each lattice among the imaging plane. For single-point convergence imaging (i.e., M=1), the phase distribution k0|R⇀uv−r⇀mn| guarantee Eq. (1) ensures that the scattering components of the U×V meta-atom structures are all in phase superposition, resulting in the maximum amplitude at the focus. However, when M>1, it cannot be guaranteed that the scattering components of each meta-atom structure at each focus are in phase. Therefore, various holographic algorithms are used to achieve the desired imaging effect based on different setting conditions and implementation approaches. Developed from the classical Gerchberg–Saxton algorithm with an optimized weighting parameter enabling both holographic imaging efficiency and the predesign intensity ratio among the desired focus spots, this desired image can be obtained effectively based on the phase-only modulation method derived without any optimization procedure, which greatly reduces the complexity of the holographic imaging design in practical applications [58,59]. Therefore, based on the phase-only modulation method, the object E-field at the AIHM plane is interference of the spherical waves originated from the multifocuses, resulting in the detail derivation as Eobj=(1,0,0)∑mM∑nNAmn|R→uv−r→mn|⁢exp(jk0|R→uv−r→mn|−jk→0·r→uv),(1)where Amn is the relative intensity referring to the weight factor shaping the energy ratio between the hot spots on the mnth lattice, which has been set to numerical values of 0 or 1, with 0 expressing the black pixel and 1 indicating the white pixel, as shown in Fig. 2, whereas k0 stands for the wavenumber in free space, k⇀0·r⇀uv is the radiation direction for the hologram, and (1,0,0) means that the polarization direction of our object wave is along the x-axis.

Moreover, in order to further understand the design principle and process in this proposed AIHM at microwave frequency, it is necessary to learn the theory of optical holography first. In terms of the AIHM, the interference pattern developed between the reference surface waves sensed by the feeding resonance among the composed metal meta-atoms and the object wave calculated from the abovementioned multifocus theory, is recorded by alternating bright and dark strips. Once the reference surface waves are excited among the prerecorded film made up by different meta-atoms again, a copy of the preset object wave can be produced through the developed film completely. Classically, for a reference wave ψref and object wave ψobj, the interference pattern is usually proportional to a term ψobjψref*. Therefore, once the reference wave is incident on the interference pattern, the following term obtains (ψobjψref*)ψref=ψobj|ψref|2,(2)namely, the reproduction of the preset object wave. In the present AIHM approach, the holographic concept in metasurface design is implemented by a 2D modulated tensor impedance surface, as shown in Fig. 3.

The anisotropy of the proposed impedance can be revealed by accounting for the slot-angle design of the AIHM consisting of periodic square meta-atom patches with diagonal rectangular slots. To further evaluate the anisotropic characteristics of the meta-atoms, a tensor meta-atom model is constructed with a period P of 2.7 mm, an adjacent gap ga, and a substrate thickness h of 1.5 mm. This model is designed on a single-layer grounded printed circuit board (F4B) with a dielectric constant of 2.2, operating at a central frequency of 21 GHz. As opposed to IIHM supporting a single TE or TM surface wave mode, the modulation of a hybrid mode surface wave can be converted into a leaky wave in free space. It is important to note that the impedances for surface waves propagating in different directions, denoted as θk, on scalar meta-atoms are uniform and defined as Zs. However, for surface waves propagating on tensor meta-atoms with diagonal slices, impedance tensors should be employed. These tensors are effectively equivalent to a scalar impedance for a given propagation direction θk. Generally, if the electric field is excited on the surface of a tensor meta-atom in the horizontal direction, the electric field will also be induced through diagonal slots in the orthogonal direction. This phenomenon is the reason for the slotted square producing a tensor impedance surface. Therefore, based on the interface boundary condition, the relationship between tensor impedance components and surface wave propagation direction is established [53]: $$\begin{gathered} kz/k0={−j(Z02−Zxy2+ZxxZyy) \\ ±[−(Z02−Zxy2+ZxxZyy)2+4Z02 \\ ×(Zyycos2θk−Zxysin2θk+Zxxsin2θk) \\ ×(Zxxcos2θk+Zxysin2θk+Zyysin2θk)]1/2} \\ ×[2Z0(Zyycos2θk−Zxysin2θk+Zxxsin2θk)]−1, \end{gathered}$$(3)where kz refers to the wavenumber in the z-axis direction, and Zxx, Zxy, and Zyy represent the three components of tensor impedance. Comprehensively considering the symmetrical design of this meta-atom and the law of energy conservation, the 2D tensor impedance Z should be a non-Hermitian matrix while satisfying reciprocity theory, requiring it to be a purely imaginary property, resulting in Zxy=Zyx. Accordingly, given the surface wave propagation direction angle θk, the equivalent normalized surface impedance kz/k0 can be determined to correspond to a specific tensor impedance matrix, forming a method to analyze the feature in tensor impedance based on the equivalent scalar impedance Z0kz/k0. Furthermore, equivalent scalar impedances along with different preset directions θk in business simulation software based on the finite integration method have been calculated as shown in Figs. 3(c) and 3(d). In terms of these simulated results and our previous research [60], it can be seen that the curve of the equivalent scalar impedance versus the variable θk within the 360° range is elliptical in shape, with its major axis angle almost coinciding with the slotted angle θs. Consequently, according to Eq. (3), for a given tensor impedance, the maximum equivalent scalar impedance Zemax and its corresponding propagation angle θkemax can be calculated simultaneously. Then, we can directly solve for the meta-atom size using the relationship curve between the meta-atom gap and the maximum equivalent scalar impedance shown in Fig. 3(b).

As mentioned above, the excitation source positioned at the origin of the coordinate system lying in the XOY plane interacts with the meta-atom interface, resulting in the generation of the reference surface current depicted by Jref=(xuv,yuv,0)e−jk0n|r⇀uv|/|r⇀uv|.(4)

In accordance with classical holographic theory, the leaky vector term Eobj|Jref|2 can be scattered through a vector surface wave modulated on a tensor impedance proportional to Z∝Eobj⊗Jref†, as proposed in Ref. [53]: Z·Jref∝Eobj·|Jref|2.(5)

Furthermore, the developed pure imaginary tensor impedance can be obtained by subtracting the Hermitian conjugate term of Eobj⊗Jref†, fully satisfying the requirements for energy conservation and reciprocity. This yields $$\begin{gathered} Z=(ZxxZxyZyxZyy) \\ =j(Xa00Xa)+jMa2Im(Eobj⊗Jref†−Jref⊗Eobj†) \\ =[jXa+jMa2Im(Exobj*Jxref†−Exobj†*Jref) \\ ⁢jMa2Im(Eyobj*Jxref†−Exobj†*Jyref) \\ ⁢jMa2Im(Exobj*Jyref†−Eyobj†*Jxref) \\ ⁢jXa+jMa2Im(Eyobj*Jyref†−Eyobj†*Jyref)], \end{gathered}$$(6)where † refers to the conjugate operator, ⊗ denotes the Kronecker product, Exobj and Eyobj are the components of the object field on the x- and y- axes, Jxref and Jyref are referenced current components along the x-axis and y-axis, and Xa and Ma denote the average reactance and modulation depth, respectively, depending on the maximum value Zemax and the minimum value Zemin among equivalent scalar impedances distributed throughout the entire AIHM. To develop the object holograms, we substitute the desired wave from Eq. (1) and the reference current from Eq. (4) into the modulation method described by Eq. (6). Consequently, the tensor impedance function with a given spatial direction angle (θs,φs) versus variable positions can be calculated as follows: {Zxx=j[Xa+xuvMa|r⇀uv|∑mM∑nNAmn|R⇀uv−r⇀mn|sin(k0|R⇀uv−r⇀mn|+k0n|r⇀uv|−k0xuvsinθrcosφr−k0yuvsinθrsinφr)]Zxy=jyuvMa2|r⇀uv|∑mM∑nNAmn|R⇀uv−r⇀mn|sin(k0|R⇀uv−r⇀mn|+k0n|r⇀uv|−k0xuvsinθrcosφr−k0yuvsinθrsinφr)Zyy=jXa.(7)

Moreover, to further validate the flexibility of the proof of concept, six AIHMs comprising 101×101 meta-atoms covering an area of 272.7mm×272.7mm have been constructed to generate a sequence of holographic images. These images include not only three Arabic numerals of “1,” “2,” “3,” but also three capital letters of “A,” “B,” “C.” Simultaneously, the hologram patterns shown in Fig. 2 have been meshed into 64×64 grids with a spacing of 7.1 mm, which is close to the half-wavelength at 21 GHz. Therefore, the tensor impedances distributed at different positions among the AIHM for reconstructing the desired holograms with a radiation angle (θs=0°,φs=0°), at a distance of 38 mm relative to the origin of O, can be calculated based on Eq. (7), as shown in Figs. 4 and 5.

For the calculated tensor impedances known in Eq. (3), the maximum equivalent scalar impedance of this AIHM hologram Zemax and its coincident propagation angle θkemax, which approximately equals the meta-atom slotted angle θs, can be directly determined. Consequently, Zemax, the gap ga, and the slotted angle are derived based on the maximum equivalent scalar curve in Fig. 3(b) using the interpolation fitting method, as illustrated in Figs. 6 and 7. Subsequently, AIHMs enabling hologram images of “1,” “2,” “3,” “A,” “B,” “C” in the near field with the radiation direction angle (θs=0°,φs=0°) are parametrically constructed based on our previously developed meta-atoms to achieve full-wave simulation stemming from finite element method (FEM). From numerical simulations shown in Figs. 7 and 8, it is evident that the holograms fed by the monopole source with the TM01 mode are effectively detected at the predesigned imaging plane at z=38mm, displaying the Arabic numerals of “1,” “2,” “3” and the capital letters of “A,” “B,” “C,” thus realizing clear and complete holograms across a wide operating frequency band from 18 GHz to 22 GHz. Furthermore, to confirm the flexibility of the beam arbitrary pointing control for our present AIHMs in hologram imaging applications, four AIHMs composed of 131×131 meta-atoms with different spatial radiation angles of (θs=10°,φs=0°), (θs=−15°,φs=0°), and (θs=20°,φs=0°) are developed to achieve the capital letter “B” based on the tensor impedance components determined in Fig. 10 and the meta-atom geometrical parameters calculated in Fig. 11. To avoid the intersection of the imaging plane and the plane where the AIHM is located, we increased the imaging distance to z=80mm; meanwhile, the resolution of the target image is also raised to 101×101 pixels with a square length of 7.1 mm to maintain the imaging effect. However, we also set the broadside angle of (θs=0°,φs=0°) as a reference for other tilt angle radiations, as shown in the first column of Fig. 12. Compared to the referenced simulation result at the angle of (θs=0°,φs=0°), all other holographic images of capital “B” working at the same frequency gradually distort as the tilt angles increase, but the main features of the preset pattern can still be clearly recognized. Thus far, the simulation results from Figs. 8, 9, and 12 have successfully demonstrated the flexibility of the proposed method in imaging pattern design and radiation direction control simultaneously.

To further experimentally demonstrate the feasibility of our present concept of AIHM holograms, 10 elaborate AIHMs mapped with the same parameters as the simulated models are fabricated using mature printed circuit board (PCB) processing technology. In these prototypes, all are equipped with SubMiniature version A (SMA) connectors serving as monopole feeding sources penetrating through the antenna, as illustrated in Fig. 13. Additionally, a near-field measurement system is established using the laser parallel plane calibration method, as depicted in Fig. 14.

For broadside holographic imaging measurements, as shown in Fig. 14(a), the probe scanning plane is set 38 mm away from the AIHMs with the radiation angle (θs=0°,φs=0°) to complete six near-field plane tests with a sampling step of 2.4 mm, capturing the horizontal polarization field Eh corresponding to the radiation field polarization. Furthermore, as illustrated in Fig. 14(b), the near-field scanning plane, located at a distance of 80 mm, is flexibly modulated by its own mechanical arm to adjust the sampling tilt angles, satisfying our preset radiation angles of (θs=10°,φs=0°), (θs=−15°,φs=0°), and (θs=20°,φs=0°), respectively. To further determine the impedance matching characteristics of the manufactured AIHMs by evaluating the reflection coefficients, which is the ratio of the amplitude of the reflected wave to the amplitude of the incident wave, we measured the S₁₁ parameter of the feed port. As depicted in Fig. 15, good impedance match is demonstrated with S11<−10dB ranging from 18 GHz to 22 GHz. The measured near-field holographic imaging results, exhibited in Figs. 16 and 17, clearly reveal the contour features of Arabic numerals “1,” “2,” “3” and capital letters “A,” “B,” “C” operating from 18 GHz to 22 GHz in the frequency domain, thereby proving the flexibility of near-field holographic imaging based on this proposed AIHM. Minor fluctuations between the simulations and the experimental results are observed, primarily attributed to limitations in processing accuracy and the stability of the dielectric constant of the medium. Moreover, from the measured tilt radiation results displayed in Fig. 18, a clear capital letter “B” has been successfully reconstructed on the imaging plane. However, as the tilt radiation angle gradually increases, the overall imaging quality of our desired pattern deteriorates, consistent with the simulated results shown in Fig. 12. The slight disagreement between the simulations in Fig. 12 and the experiments in Fig. 18 can also be attributed to position errors in the tilt scanning plane. Nonetheless, this does not affect the measured results that demonstrate the degree of freedom in regulating the imaging radiation angle. Hence, the relative bandwidth of our proposed method based on these AIHMs for holographic imaging is at least 19%, providing high imaging quality, thereby achieving the goal of surface-wave modulated holography with good performance.

In summary, the concept of holographic imaging based on AIHMs for reconstructing various capital letters and Arabic numerals operating from 18 GHz to 22 GHz has been developed, primarily focusing on broadside radiation. Clear images can also be easily distinguished within a 20° tilt radiation angle, effectively demonstrating the flexibilities in near-field imaging and radiation direction control. Moreover, a synthetic approach combining near-field focusing and holographic leaky-wave methods applied to microwave holography has been significantly validated through the good consistency between the measurements and theoretical analyses. Furthermore, compared to prevalent space-wave modulated methods in microwave holography, the proposed surface wave technique offers a different pathway to holographic imaging, leveraging inherent advantages such as system integration, low processing costs, and design convenience.

Acknowledgment. This work was supported by the China Postdoctoral Science Foundation, the National Natural Science Foundation of China, the Defense Program Projection and the Foundation of National Key Laboratory of Radar Signal Processing.