Millimeter wave antennas have been widely applied in the advanced wireless communication systems for wideband and large-channel-capacity applications¹. The compact sizes enable millimeter wave antennas to be integrated with radio frequency integrated circuits (RFIC) for reducing the loss introduced by interconnections^2⇓-4. To deal with the intrinsic high path attenuation in the atmosphere, antenna arrays with high gain and beam forming abilities are required for various millimeter-wave applications⁵. The phased array is an optimum choice for beam forming and synthesizing^{6⇓⇓⇓⇓-11}. Conventionally, feeding networks are used for a particular beam by assigning uniform amplitude and phase to each element^12⇓-14. In addition to the uniformly distributed aperture, the tapered amplitude distributions are also used for sidelobe reduction of antenna arrays. A parallel feeding network is essential for assigning different amplitude distributions including binary and Taylor distribution^15,16. Apart from the parallel feeding network, the serial feeding network can also reduce the structural complexity so as to achieve the purpose of low loss. Serial-fed patch antenna arrays^17⇓-19 and comb-line structures^19⇓-21 are usually adopted to realize linear antenna arrays with low sidelobe-levels (SLLs) by carefully designing the length and width of each segment.

Another key challenge for millimeter wave antennas is the high fabricating precision. The printed circuit board (PCB) technology is most widely used to fabricate millimeter wave antennas for the merit of low cost. However, PCB technology suffers from a moderately low precision. In millimeter wave frequencies, the permittivity of the substrate is unstable while the fabrication tolerance is relatively large compared with the operating wavelength^22,23. Therefore, it is difficult for a complex feeding network to generate an accurate phase and amplitude distribution. Another technology for millimeter wave antenna is low temperature cofired ceramics (LTCC) which is endowed with the merits of high fabricating precision and compatibility for complex structures with dozens of layers^24⇓⇓-27. The LTCC technology provides a pathway to fabricating feeding networks with more complex layered structures and higher dielectric loss, especially for a large-scale feeding network. Air-filled waveguides are the optimum transmission structure for eliminating dielectric losses. An advanced technology, which is named as bulk silicon microelectromechanical systems (MEMS) micromachining, is developed for designing and precisely fabricating millimeter wave and terahertz antennas with low dielectric loss and high radiation efficiencies^{28⇓⇓⇓-32}. With the introduction of this technology, air-filled type of antennas could be easily fabricated and the dielectric loss is completely avoided. Unlike the LTCC technology, bulk silicon MEMS micromachining is not compatible for too many layers, consequently an antenna array solution with reduced structural complexity is demanded.

The recent research progress in metamaterials provided another approach to solving the problems mentioned above. The epsilon-near-zero (ENZ) medium, which is of a low permittivity and an infinite wavelength, has been proved to exhibit a series of exotic geometry-independent phenomena for deformable cavities, antennas, and other microwave components^{33⇓⇓⇓⇓-38}. For instance, the supercoupling effect of ENZ medium^34,35 illustrates that electromagnetic waves could be totally transmitted through the extremely deformed tunnels even with sharp angles filled with ENZ medium. In contrast to the conventional resonators whose resonant frequency varies obviously when its geometry changes³⁶, the resonant frequency of ENZ-filled resonators tends to be geometry-independent. In addition, ENZ metamaterials have also been utilized for building nanocircuits and sub-wavelength scaled computing units in optical frequencies^39,40. Moreover, innovative researches have also been completed on investigating the new physics in ENZ media. A significant example is the photonic doping of ENZ medium^41-43, which shows that the effective permeability of a bulk ENZ material can be tailored by inserting a dielectric inclusion at a single point. Another one is the analogy between ENZ medium and the ideal fluid, which theoretically and experimentally demonstrate that the Poynting vector within ENZ medium resembles the flowing of the ideal fluid^44,45. The underlying physics of both phenomena is the space-time decoupling in the ENZ medium, indicating that the spatial variation of electromagnetic fields is suppressed within this medium. To be specific, under TM incidence, the curl of magnetic fields expressed as ∇ × H = jωεE is zero in ENZ medium. In addition, the divergence of magnetic fields ∇⋅H also is equal to zero according to the Maxwell's equations. As a result, a homogeneous magnetic field is excited³³. This feature is adopted to design transmission lines without obvious phase changes⁴⁶. As a result, ENZ medium is a promising solution for applications where flexible geometry and uniformly distributed electromagnetic fields are essentially demanded.

A recent work reported that an antenna filled with ENZ medium exhibits a geometry-independent resonant frequency together with a uniformly distributed amplitude and phase on the antenna's radiating aperture⁴⁷. This feature encourages the current work to excite arbitrarily spaced antenna elements with the same amplitude and phase using ENZ medium instead of feeding networks. The homogeneity in amplitude and phase of all the elements indicates a high broadside gain. Furthermore, any broadside beams can be synthesized by optimizing the elements’ positions using the algorithms proposed by literatures⁴⁸. In this paper, a feasible method of on-chip feeding network design for large-scale antenna arrays was proposed to generate arbitrary broadside beams based on the concept of ENZ medium. While keeping the phases the constant, the positions and amplitudes of the elements were optimized in accordance with the feeding network so as to radiate a particular beam at the broadside direction. To excite such in-phase distribution, a waveguide at its cutoff frequency was utilized, several slots were etched into it for radiation. This cutoff waveguide is equivalent to an ENZ medium⁴⁹. According to the previous work⁴⁷, due to the geometry-independent property of the ENZ medium, the slots’ positions were arbitrarily arranged along the waveguide without influences on phase distribution or impedance matching. Furthermore, the amplitude of each element was controlled by the width of the slot. For high precision fabrication, the bulk silicon MEMS micromachining technology was adopted for fabricating a hollow waveguide working at its cutoff frequency to emulate the ENZ medium, on which radiating slots were etched at the optimized positions. The fabricated antenna array exhibits a high gain of 13.6 dBi with a SLL of −20.0 dB. Moreover, this ENZ-based feeding structure is also feasible for feeding a uniform aperture and quasi-nondiffractive beams, demonstrating the potential of the proposed method.

The general working mechanism is depicted in Fig. 1a. It contains a thin cavity filled with ENZ medium serving as the feeding network. 2N identical slots are etched on this cavity for radiation and a specific example is presented, in which 2N = 4. As has been introduced above, the magnetic field within the cavity is a constant H₀. Since all the slots are identical, the input impedances of these slots are the same, so that E_n/H_n is a constant, demonstrating that E_n is a constant because H_n= H₀. As a result, all the slots are excited with equal amplitude and phase which is independent to their spacings. To further investigate the mode within the ENZ feeding network, the Faraday's Law can be applied to the cavity:

Here, d₀ is the width of the feeding waveguide, while d_n is the width of n-th slot. E_inc, E_r, and E_n represent incident, reflected and each element's electric field, respectively. It is assumed that the thickness h is small enough to be neglected, so that the following equation is obtained:

After that, the Faraday's Law was applied to each section of the cavity. For positions where x₂ < x < x₁, the following equation was obtained:

Since E₁d₁ >> jωµ₀h(l/2-x)H₀, E(x) = E₁d₁/h applies to all x subject to x₂ < x < x₁, approximately. Similarly, E(x) = 2E₁d₁/h applies to all x subject to 0 < x < x₂ and E(x) = 0 applies to x > x₁. This derivation is also valid for a general case with 2N elements, while the following equation is valid for this general one:

Here, V_s = d₀(E_inc+E_r). In the x<0 region, E(x) = −E(−x), indicating that the electric field is odd symmetric. As shown in Fig. 1b, the stair-like electric field distribution contributes to the uniform-amplitude and in-phase feeding of these elements with increased degrees of freedom on spacings.

On the other hand, as demonstrated in the previous work⁴⁷, such an antenna is equivalent to a lumped circuit in series with an inductor whose inductance is proportional to the cross section of the cavity. The Kirchhoff's law describing the voltage and current of this circuit is written as:

Here, V_s denotes the voltage of the source and V_n the voltage on each slot. I₀ is the equivalent current of the circuit and is quantitatively equal to H₀. N is half of the elements’ number. L_ENZ is the serial inductor introduced by the ENZ medium according to ref.⁴⁷ and is proportional to the cross-sectional area of the ENZ cavity. In particular, as h approaches zero, the proposed waveguide structure exhibits a thin profile so that the additional inductance is small enough to be negligible. Therefore, the equivalent circuit model of this antenna is obtained as shown in Fig. 1c, which turns out to be a simple lumped serial circuit in which each radiation slot i is modelled as a resistor R_i. According to this circuit model, these identical elements share the same amplitude and phase, which is independent to their positions and spacings. As a result, the proposed ENZ waveguide is capable of feeding an arbitrarily spaced antenna array to each element with the same amplitude and phase. In addition, this circuit model also yields a stair-like voltage distribution in x > 0 region described by：

Here, x_n is the position of the n-th element. Odd symmetry also applies to this circuit. The voltage distribution of the equivalent circuit is shown in Fig. 1d, which resembles Fig. 1b well.

Based on the working mechanism discussed above, an on-chip antenna array was developed. The general concept of the antenna array is depicted in Fig. 2a. The input RF power is fed into the ENZ waveguide through the feeding slot and radiates to the free space in the broadside direction with a reduced sidelobe level. As shown in Fig. 2a and 2b, the whole structure is laminated by three gold-plated silicon layers. On Layer 1, ten identical slots are etched with optimized spacings to achieve such sidelobe reduction. An air-filled cavity is formed by etching a rectangular region in Layer 2 just beneath Layer 1. The width of this cavity is approximately half the wavelength so that it is equivalent to a waveguide filled with ENZ medium⁴⁹. Layer 3 is on the bottom of the antenna and an H-shaped slot is etched as the feeding slot which allows electromagnetic power coupled with the input port. All the electromagnetic power is limited within the air-filled structure by gold plates, while the lossy silicon introduces no dielectric loss, which enables a high radiation efficiency of this antenna array. For the convenience of measurements, fixing holes are drilled around the antenna array for connection with WR-15 waveguide. This antenna array is fabricated via silicon MEMS micromachining process^28-32 and the photos of the prototype are shown in Fig. 2c. The detailed fabrication process is illustrated in the part of Methods.

To further validate the mechanism, numerical simulations were launched to investigate the field distributions within the waveguide. The simulated z component of magnetic field on the y=0 plane is depicted in Fig. 3a, from which it can be observed that the magnetic field maintains a uniform distribution within a length of 3.4λ₀ (λ₀ is the free-space wavelength) due to the stretched wavelength within ENZ medium. According to Fig. 3a, the magnetic fields on each slot are also with the same amplitude and phase, which demonstrates that the ENZ waveguide excites a uniform amplitude and phase distribution. As shown in Fig. 3b, the simulated electric field within the ENZ waveguide exhibits an exotic stair-like distribution. To be specific, the electric field is a constant between two adjacent slots while it experiences a sharp variation at the slot. This stair-like distribution is consistent with the results obtained by Eq. (4). Moreover, the amplitudes of electric fields on each slot are analyzed quantitatively. Fig. 3c and 3d plot the amplitudes and phases of the electric fields on the aperture. The electric fields are almost the same on every slot except that the amplitude slightly decays along the waveguide. Nevertheless, all the slots can be viewed as sharing approximately the same amplitude and phase.

Based on the feeding structure mentioned above, the low-sidelobe array is synthesized by numerical optimizations. Using the numerical optimization algorithm⁴⁸, a 10-element array with uniform amplitude and phase has been synthesized by optimizing the spacings between the elements. The array factor is the radiation pattern when all the elements are radiating isotopically derived as:

Here, k₀ is the wavenumber in freespace and x_m is the position of each element. It is assumed that the amplitude and phase of each element is the same while achieving a low SLL by optimizing x_m. The optimized result of SLL is calculated to be −20.62 dB within a total length of 3.0 λ₀, in which 10 elements are used. The minimum value elements’ spacing is selected to be 0.15 λ₀ so as to avoid too strong mutual couplings between adjacent elements, which would disturb the amplitude and phase distribution.

The measurement results of fabricated antenna prototype are shown in Fig. 3e-i in comparison with the simulation results. The reflection coefficients were firstly measured using a vector network analyzer and depicted in Fig. 3e together with the simulated ones. When the reflection coefficients are lower than −10 dB, the impedance bandwidth was simulated to be 58.95-60.45 GHz (2.51%) and measured to be 58.75-60.28 GHz (2.57%). Moreover, the radiation performances of this antenna array were also evaluated through measurements in an anechoic chamber, while the measurement setup is shown in the part of Methods. The simulated radiation patterns and measured patterns from −60° to +60° are both depicted in Fig. 3f. The broadside gain of antenna is 14.1 dBi by simulation and 13.6 dBi according to the measurements. In addition, the radiation efficiency is also evaluated and shown in Fig. 3g. Due to the measurement limitations, the efficiency of this antenna prototype cannot be directly measured. Instead, the radiation efficiency was evaluated by the ratio between measured gain and simulated directivity. Under this manner, the measured efficiency might be slightly lower than its real value since not only the losses (metal loss, dielectric loss and reflection loss) are taken into consideration, while the leakage of the connection structure between waveguide and antenna results in an unprecedented high backward radiation, which therefore deteriorates the both measured gain and measured efficiency. The simulated and measured efficiencies are 97.6% and 84.8%, respectively. The radiation patterns in both x-z plane and y-z plane at 60.0 GHz are depicted in Fig. 3h and 3i. This antenna radiates a directive pattern in x-z plane and a wide beam in y-z plane in the broadside (z = 0) direction. According to Fig. 3h, the simulated SLL of this antenna array is −22.3 dB, while the measured result is −20.0 dB. All the measurement results agree well with the numerical simulation ones.

To further evaluate the performances, a comparison has been made between the proposed design and other state-of-art low-SLL millimeter wave antenna arrays. Above all, the proposed antenna array achieves the highest efficiency according to simulation, while the measured efficiency of 84.8% is also much higher than others. Together with a moderate bandwidth of 2.57% and comparable SLL of −20.0 dB, the proposed antenna realizes a high gain of 13.6 dBi within such a small length of 3.4 λ₀, demonstrating that a high aperture efficiency is obtained. Although higher gains may be achieved by some of the previous antenna arrays^18,51, their sizes are much longer than those of the proposed ones. This improvement in aperture efficiency is resulted from the feeding structure based on low-loss ENZ and the high uniformity of phase distribution on the aperture guaranteed by the intrinsic uniform field distribution of the ENZ medium, which validates the advantages of the proposed method in achieving high aperture efficiency Table 1.

In addition to the sidelobe reduction that has been discussed above, the proposed ENZ feeding network is also useful for synthesizing other beams. As a specific example, the maximum main lobe antenna array is realized when all the identical slots are spaced with equal spacing of 1.67 mm (0.833 λ₀). In this case, this antenna array reaches its highest gain. The simulation results of this high gain antenna are shown in Fig. 4. Similar to the low SLL case, the electric field depicted in Fig. 4a and magnetic field in Fig. 4b of such high gain antenna are all consistent with the circuit model and validate that all the elements are with the same amplitude and phase according to Fig. 4c and 4d. Furthermore, the reflection coefficients, radiation efficiencies and radiation patterns are also simulated and shown in Fig. 4e-g respectively. This antenna is simulated to exhibit a −10-dB impedance bandwidth of 58.5-60.3 GHz and a peak gain of 14.6 dBi. The peak radiation efficiency is simulated to be 98.0%.

Another example is generating a nondiffractive Bessel beam with the proposed ENZ feeding structure. It is well-known that the Bessel beam maintains a nondiffractive wavefront in near field^52,53. A planar Bessel beam yields a near-field distribution subject to the Bessel function A_i = J₀(k_cr_i), where A_i is theamplitude of the i-th element located at r_i and J₀() is the zeroth-order Bessel function. Similar to the ordinarily defined Bessel beam, a quasi-nondiffractive Bessel beam is realized using the proposed ENZ antenna array, which is featured by a fan beam whose radiation power is constrained in one dimension while diffracted in the other. To form the amplitude distribution governed by Bessel function, a fine tuning of width of the slots is needed. In particular, the configuration together with the detailed dimensions of the aperture layer (Layer 1) is shown in Fig. 5a.

The performances of this antenna were also numerically investigated. The electromagnetic field distributions within the ENZ waveguide are shown in Fig. 5b and 5c, and they agree well with the theoretical results. The electric field on the aperture is also depicted in Fig. 5d and 5e, which validates that an in-phase distribution is excited while the amplitude is tuned by the width of the slots. The reflection coefficient, radiation efficiency, and radiation patterns in the x-z plane are shown in Fig. 5f-h, respectively. This antenna is simulated to exhibit a −10-dB impedance bandwidth of 58.7-60.0 GHz as shown in Fig. 5f and a SLL of −28.5 dB. The peak radiation efficiency is simulated to be 97%. Moreover, the near-field electric field distribution is depicted in Fig. 5i, from which it can be observed that the beam maintains the same profile in x-z plane while diffracts in y-z plane, featuring a fan-shaped characteristic. All these results validate the feasibility of the proposed ENZ feeding network for on-chip antenna array.

In the current work, an ENZ-based feeding network was proposed for low-loss beam synthesis applications of on-chip antenna arrays. Through both the equivalent circuit model and numerical simulations, a new stair-like mode in ENZ medium-filled waveguide has been discovered, which demonstrates that such ENZ waveguide is feasible to assign the same excitation phase to each element. Based on this result, a low sidelobe antenna array was further synthesized by optimizing the positions of ten identical radiating slots with the same amplitude and phase etched on the ENZ waveguide. With the bulk silicon MEMS micromachining technology, such an antenna array was precisely fabricated operating at millimeter wave frequencies and easily integrated with RFICs. Within a geometry size of 0.5 × 3.4λ₀², this antenna array was tested to achieve a gain of 13.6 dBi and a SLL of −20.0 dB at 60.0 GHz within the −10-dB impedance bandwidth of 2.57%, which exhibites both higher radiating efficiency and aperture efficiency than previous works. Moreover, a maximum-mainlobe beam and a quasi-nondiffractive Bessel beam were also successfully designed based on the proposed antenna array feeding by ENZ waveguides, demonstrating that the proposed method is a promising on-chip solution for synthesizing arbitrary beams at the broadside direction.

Numerical simulations The numerical simulations on the 3D structure have been carried out with ANSYS HFSS® 18. During the simulations, the materials of the wafers are asserted to be gold rather than gold coated silicon, since the thickness of the gold plating is much thicker than the skin depth, therefore it is assumed that there is no electromagnetic field in the silicon region. A wave port with a height of 3.88 mm and a width of 1.76 mm was used to simulate the WR-15 waveguide excitation. Only the fundamental mode of this wave port is taken into consideration. Radiation boundaries are assigned with a 5-mm distance to the antenna in all directions. The near field distribution of the quasi-nondiffractive beam was simulated using CST Microwave Studio 2016. The simulation settings are the same with that in ANSYS HFSS® 18 as mentioned above except for adding another air box. In particular, an additional air box was added above the antenna in z-direction with the size of 30×30×60 mm³. The radiation boundaries are assigned with a distance of 5 mm (1.0λ₀) away from the antenna structure.

Fabrication The fabricating process of the antenna array proposed in this paper is similar to the bulk silicon micromachining process reported in (26-30). A flow chart of the Si-MEMS micromachining fabricating process is shown in Fig. 6. The total process was divided into several steps. At the first step, a silicon wafer was covered by a mask layer on its surface. In the second step, the photolithographing technique was adopted to etch the mask on the surface of the wafer. The mask on the region designed to be air-filled is etched by laser. Subsequently, the wafer was exposed in sulfur hexafluoride as the etching gas for deeply etching. Different from the conventional CMOS technology, the wafer was etched through by the etching gas and the silicon not covered by the mask was completely removed in this step, and air-filled structures were thus formed after this process. The etching precision is 4 µm and a chamfer with the radius of 50 µm is at the corner. During the next steps, the masks were cleaned and gold layers were plated on all the surfaces and sidewalls of the silicon wafer. The thickness of the gold layer is 3.5 µm with a tolerance of 0.5 µm. During this process, the silicon wafer was turned into a fully-metallic component in terms of electromagnetic properties and the lossy silicon dielectric was not illuminated by electromagnetic waves. As a result, the fabricated antenna exhibits near-zero dielectric loss. Finally, three wafers processed by the steps mentioned above were bonded together at the final step. The wafers were firstly optically aligned with a tolerance of 5 µm and then directly bonded together based on the interatomic interactions without any additional layers. A three-layered all-metallic structure filled by air was formed when adopting this bulk silicon MEMS fabricating technology, which is endowed with the merits of high precision and low loss, and is essential for millimeter wave antenna applications. The fabricated antenna is also compatible to be integrated with RFIC by either directly bonding or flip bonding technology.

Antenna measurements The S-parameters of the antenna was measured by adopting vector network analyzer N5227A. The antenna prototype under tested was fed by WR-15 rectangular waveguide. A waveguide-to-coaxial converter was used for inter-connection between the WR-15 waveguide and 1.85-mm coaxial cable, which is connected to the vector network analyzer. The radiation patterns and gains were measured in an anechoic chamber. The measuring setup is shown in Fig. 7, where the near-field scanning technique is utilized. The radiation pattern was retrieved by Fourier transmission based on the near filed measurements. When measuring the gain of the antenna, a standard gain horn antenna operating from 50 GHz to 75 GHz was utilized for calibration. Instead of being measured directly in anechoic chamber, the measured efficiency was calculated through dividing the measured gain by the simulated directivity. In this case, the leakage of the feeding structures and the fabricating tolerances also lead to the reduction in efficiency, which is mainly caused by Ohmic losses as the former two factors deteriorate the gain.

Acknowledgments The authors would like to express their sincere gratitude to Foshan University (Foshan, China) for their assistance on antenna measurements.

Data availability All the data needed to evaluate the conclusions of the current work are present in the paper. Additional data related to this paper may be requested from Y.L.

Funding This work was supported by National Natural Science Foun- dation of China (NSFC) under grant U22B2016 and 62022045, the Na- tional Key Research and Development Program of China under Grant 2021YFA0716600, and the Shenzhen Science and Technology Program un- der Grants JSGG20210802153800002.

Declaration of Competing Interest The authors declare no competing interests.