Active Hemispherical Metasurface Transmitarray Antenna for Wide-Angle 3D Beam Steering and Target Tracking

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Active Hemispherical Metasurface Transmitarray Antenna for Wide-Angle 3D Beam Steering and Target Tracking

arXiv:2608.08991v1 [cs.NI] 10 Aug 2026

Somayeh Komeylian, Member, IEEE and Christopher Paolini, Member, IEEE

Abstract—A stacked multilayer hemispherical graphene-based transmitarray antenna operating at 250 GHz is developed to achieve wide-angle 3D beam steering. This work presents the first fully automated CST Studio Suite macro for constructing the multilayer hemispherical geometry and configuring 121 independently tunable graphene sectors. Numerical results confirm that the integration of voltagecontrolled graphene sectors with FCC gold patches enables wideangle beam steering, achieving an elevation scanning range of −78◦ to 78◦ in θ and full 360◦ azimuthal coverage while maintaining stable radiation characteristics and broadband impedance matching (|S11 | < −10 dB) with a relative bandwidth of 40%. The antenna achieves a directivity range of 11.38–14.72 dBi and an SLL range of −6.8 to −9.8 dB across all investigated steering directions. It also maintains high antenna efficiencies from 71% to 80% with a 5% degradation due to imperfections across the entire scanning range, making it a promising candidate in practical moving-target-tracking applications. Index Terms—Graphene surface impedance, metasurfaces, transmitarray antennas, beam steering and reconfigurable antennas.

I. I NTRODUCTION CTIVE reconfigurable beam-steering architectures have been extensively investigated by several research groups, enabling dynamic control of electromagnetic wavefronts through tunable metasurface elements, Tables I and II. During the early development of graphene-based tunability concepts between 2015 and 2018, several research groups [1], [2], employed reflectors to demonstrate the feasibility of grapheneenabled electromagnetic reconfigurability. In this sense, when a plane wave impinges on a surface, the resulting scattered electromagnetic fields are determined by solving Maxwell’s equations subject to the appropriate boundary conditions [3], [4]. For a non-uniform surface, the assumptions underlying Snell’s law are not satisfied, as the reflected field is generated by a spatially varying impedance distribution rather than a homogeneous planar interface [5]. Consequently, the electromagnetic response should be determined using full-wave scattering analysis [6], [7], [4]; however, such approaches are computationally expensive for complex multilayer and electrically large reflectors.

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S. Komeylian is with the Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, CA 92093 USA, and the Department of Electrical and Computer Engineering, San Diego State University, San Diego, CA 92182 USA (e-mail: skomeylian@ucsd.edu). C. Paolini is with the Department of Electrical and Computer Engineering, San Diego State University, San Diego, CA 92182 USA (e-mail: paolini@engineering.sdsu.edu).

Alternatively, when the surface is modeled as a twodimensional periodic structure, the complexity of the electromagnetic scattering problem is significantly reduced, enabling efficient analysis of the unit-cell response under periodic boundary conditions. By exploiting the Floquet–Bloch theorem [8] the governing wave equation within the periodic medium is transformed into a closed-form secular equation. This substantial reduction in computational complexity has established periodic reflectors as an effective framework for analyzing and designing a wide range of electromagnetic systems [9], [10], [11], enabling efficient extraction of the reflection coefficient S11 without the need for full-space bruteforce scattering analysis. As an emerging class of tunable reconfigurable reflectors, graphene-based reconfigurable intelligent surfaces (RISs) [12], [13], have attracted significant attention as a promising technology for future 6G networks due to their tunability, compact implementation, particularly because of the aforementioned simplicity in modeling its electromagnetic response through periodic unit-cell analysis. However, practical RIS deployment architectures, channel models, and integration into cellular systems remain active research areas, since 6G standardization is still in progress and RIS has not yet been adopted as a standardized component in 3GPP specifications [14], [13]. RIS technology has attracted substantial attention as a promising solution for scenarios where direct antenna coverage is compromised by severe blockages, challenging propagation environments, or the prohibitive cost of deploying additional network infrastructure [15], [16], [17]. In particular, graphene-based RIS solutions offer a viable approach for mitigating non-line-of-sight (NLoS) communication limitations caused by dense obstacles while providing a compact and potentially cost-effective alternative to conventional basestation or relay deployment in challenging environments. However, conventional RIS configurations inherently rely on the incident field distribution, causing the reflected wavefront to remain coupled to the spatial amplitude and phase variations of the incoming wave across the aperture. This inherent coupling restricts independent wavefront manipulation, particularly under non-uniform illumination, where spatial variations in the incident field can degrade beamforming performance [18], [19], [20]. Unlike conventional reflective RIS architectures, which are mainly designed to manipulate wireless propagation channels and enhance coverage through reflected wave control rather than direct beam synthesis, our proposed hemispherical transmitarray antenna, Fig. 1, achieves agile electronic beam steering with

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robust real-time tracking capability in dynamic environments tenna produces electronically reconfigurable far-field radiation [18], [15], [13]. patterns for agile beam steering. To overcome these limitations, our proposed hemispherical Another significant advantage of our proposed hemispherical antenna facilitates full radiation pattern tailoring through transmitarray antenna over conventional planar transmitarray voltage-controlled graphene-driven beam steering [21], [22]. antennas and reflectors is the intrinsic alignment between the In contrast to conventional RIS architectures, our transmitarray incident wave vector and the local surface normal, which antenna further provides a control over the elevation, azimuth, substantially reduces the feed-induced phase compensation steering angles, and the HPBW. Specifically, in the proposed required by individual unit cells. Considering the general transmitarray antenna, the circular feed horn and hemispherical monochromatic phase term e−jK·R , the propagation characlens determine the HPBW, whereas the bias voltages applied to teristics of an electromagnetic wave in planar and lens-based the graphene sectors independently regulate the beam-steering configurations can be described by the spatial relationship direction. between the propagation vector K and the position vector R. Dynamic radiation-pattern reconfiguration is essential for Typically, a horn antenna employed as a feed source for either tracking moving targets in dynamic environments. By in- planar [24], [25] or curved [26], [27] configurations illuminates tegrating real-time sensing, trajectory prediction, adaptive their corresponding apertures with spherical wavefronts, whose positioning, and closed-loop control, a robust tracking system spatial propagation is characterized by the wave vector K. can continuously steer the antenna beam to maintain a high-gain When spherical wavefronts illuminate a planar configuracommunication link along the target trajectory. tion [24], [28], the unit cells distributed across the aperture In the proposed architecture, the bias voltages applied experience nonuniform incident phases due to their different to the graphene sectors can be potentially generated by a path lengths from the feed source. This feed-induced phase high-speed FPGA-based control system, enabling real-time variation appears as an additional phase term in the array electronic beam steering through fast spatial reconfiguration factor, requiring accurate compensation during aperture phase and agile beam deflection. Consequently, the hemispherical synthesis to achieve the desired radiation pattern. In other words, transmitarray antenna can continuously align its main beam the projection of the propagation vector K onto the position with moving targets while preserving high directivity and vector R of each unit cell results in an additional spatial reliable communication links. phase contribution, which should be accounted for in the array The necessity of dynamically reconfiguring an antenna factor formulation. In contrast, our proposed hemispherical lens radiation pattern for moving-target tracking stems primarily configuration places the feedhorn phase center at the center from three core operational requirements: (1) beam steering or of curvature, ensuring that the propagation vector of K is directional tracking: our proposed hemispherical transmitarray approximately aligned with the radius vector R at every point antenna dynamically aligns its main radiation beam with across the lens aperture. Because both vectors are perpendicular the target’s time-varying direction of arrival to sustain a to the local surface, the FCC unit cells experience nearly stable, high-gain communication link. This rapid beam-steering uniform incident phases, thereby inherently minimizing feedcapability is achieved by dynamically controlling the bias induced spatial phase variations across the aperture. As a result, voltages applied to the graphene sectors, thereby enabling the feed-induced spatial phase variation that arises in planar adaptive wavefront synthesis through precise modulation of configurations [24], [25], [28], is substantially mitigated in the aperture phase distribution, (2) beamwidth adaptation or our proposed hemispherical antenna, owing to the intrinsic HPBW control: the hemispherical lens and feedhorn enable alignment between the incident propagation vector and the local adaptive HPBW control, providing a broad beam for efficient surface normal across the curved aperture. Consequently, the initial target acquisition and a narrow, high-gain beam for intrinsic suppression of feed-induced spatial phase variations lock-on tracking to maximize the received signal strength and in our proposed transmitarray antenna reduces the complexity signal-to-noise ratio (SNR), and (3) sidelobe and interference of aperture phase synthesis, improves aperture efficiency and suppression: dynamic radiation-pattern reconfiguration allows beamforming accuracy, and enables more efficient voltagethe hemispherical transmitarray antenna to actively suppress controlled beam steering than conventional planar transmitarray sidelobe levels and steer nulls toward multipath interference architectures. or intentional jamming sources. This adaptive spatial filtering To further highlight the functionality and advantages of significantly improves communication robustness and target- the proposed transmitarray antenna, the following discussion tracking reliability. presents the differences between reflective metasurfaces and Moreover, in contrast to available reflective RISs [19], transmitarrays. In active beam-steering systems, reflective [23], [17], the proposed antenna functions as a transmissive metasurfaces, such as RISs, can achieve wide angular scanning programmable metasurface lens antenna, where the transmitted ranges through programmable control of the reflected wavewavefront is dynamically controlled to achieve electronic beam front; however, their performance remains inherently coupled steering. to the incident field distribution. This behavior results from the The incident field generated by the feedhorn propagates S11 -based beamforming mechanism, in which the incident through the hemispherical multilayer structure and interacts electromagnetic wave interacts primarily with the surface with the tunable graphene-based unit cells and metallic res- impedance, eliminating the additional phase variations associonators, where controlled phase modulation enables adaptive ated with propagation through multilayer transmission strucwavefront synthesis. As a result, the proposed transmitarray an- tures. Consequently, the electromagnetic interactions among the

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TABLE I: Comparison of reflectarray beam-steering mechanisms based on S11 phase response and scanning range.

TABLE II: Comparison of transmissive beam-steering antennas based on S21 phase response and scanning capability.

Ref.

Freq.

Reflector Type

Ref.

Freq.

Antenna Type

[29]

125 GHz

Graphene reflectarray with an aperture diameter of 15λ

[34]

5.75 GHz

[30]

0.27 THz

Conformal metasurface lens antenna with an aperture size of ≈ 7.9λ×4.1λ

[35]

610– 685 GHz

[36]

24– 40 GHz

[37]

450– 650 GHz

[38]

2.1 GHz

[39]

115 GHz

Planar Fresnel lens antenna with an aperture size of ≈ 13.17λ× 13.17λ Flat dielectric lens antenna (FDL antenna) with a normalized aperture size of λD = 11.1, 0 where D ≈ 104.1 mm Transmit lens antenna array with an aperture size of ≈ 28.3λ Microstrip Arrayed Lines Lens (MALL) with an aperture size of 0.48λ × 0.52λ Transmissive liquid-crystal metasurface with an aperture size of ≈ 27.25λ× 27.25λ Hemispherical metasurface transmitarray antenna with an aperture diameter of ≈ 4.17λ

[31]

1 THz

[32]

14 GHz

[33]

∼1 THz

This 250 WORK GHz

Beam-Reconfiguration Mechanism

Beam Scanning Range ±45◦

Graphene chemical-potential tuning through DC biasing dynamically controls the reflection phase of resonant graphene elements. Graphene RIS Chemical-potential0◦ –40◦ with an aperture controlled graphene size of 13.6λ × conductivity provides 13.6λ discrete reflection phase states for programmable beam steering. Graphene leaky- DC bias tuning of graphene ≈ ±30◦ wave antenna. conductivity modifies the The aperture surface reactance and condimensions are trols the beam direction. not explicitly specified Mechanical re- Flexible patch elements on a ±21◦ flectarray with plastic substrate are mechanan aperture size ically deformed to modify of 12λ × 12λ the phase distribution. Graphene Chemical-potential tuning ≈ ±60◦ digital creates programmable remetasurface. flection phase gradients for The aperture beam steering. dimensions are not explicitly specified Graphene Voltage-controlled Elevation metasurface graphene chemical- angle: transmitarray potential tuning modifies ±78◦ , with an local surface impedance and aperture and transmission phase of full azdiameter of FCC resonant unit cells. imuthal ≈ 4.17λ coverage: 360◦

Beam-Reconfiguration Mechanism

Beam Scanning Range DC bias voltage tuning ±60◦ of varactor-loaded metasurface elements enables programmable transmission-phase control. Multiple feed sources further extend the steering range. Passive frequency-dependent −30◦ to beam steering using a Fresnel −14◦ zone planar lens combined with a graded-index silicon perforated disk. Passive multi-feed beam ±33◦ steering using a nine-element waveguide feed network positioned along the focal arc of a flat dielectric lens.

Mechanical translation of a ±25◦ silicon transmit lens array using a piezo actuator to modify the relative phase distribution. Passive beam steering ±60◦ using an interdigitalloaded Microstrip Arrayed Lines Lens (MALL) with controlled transmission-line phase delay. Electrical control of liquid- ±30◦ crystal molecular orientation dynamically tunes the phase/amplitude response of subwavelength unit cells.

surface elements remain relatively weak, allowing larger phase This 250 Voltage-controlled Elevation gradients to be achieved and, consequently, enabling wider WORK GHz graphene chemical angle: beam-scanning angles. Nevertheless, reflective architectures potential tuning modifies ±78◦ , are primarily suited for laboratory demonstrations, proof-ofthe local surface impedance and and transmission phase of full azconcept implementations, or applications requiring the tracking FCC resonant unit cells. imuthal of a limited number of targets. coverage: In contrast, transmissive metasurface antennas are governed 360◦ by the transmission coefficient S21 , requiring the incident electromagnetic wave to propagate through multiple dielectric and conductive layers, where inter-layer coupling and propaga- tenna efficiency, more efficient utilization of the incident tion effects collectively determine the transmitted phase and electromagnetic power, and full-space transmission, making amplitude response. These internal electromagnetic interactions, them particularly suitable for high-capacity wireless commuincluding multiple reflections, transmission losses, impedance nication systems. These characteristics render transmitarray mismatches, and interlayer coupling, severely restrict the antennas ideally suited for emerging 6G wireless networks achievable transmission phase range, amplitude response, and and broadband satellite constellations, such as Starlink, where overall beamforming performance. Consequently, owing to the high antenna gain, efficient power utilization, and reliable additional propagation and multilayer electromagnetic inter- communication links are often prioritized over maximizing the actions, transmissive metasurface antennas generally achieve beam-scanning range. Although reflective metasurfaces achieve narrower beam-scanning ranges than reflective metasurfaces, wider scanning angles, transmitarrays offer a more viable realdespite offering greater flexibility for wavefront synthesis world trade-off between beam-steering flexibility, gain, and and radiation control. Despite this limitation, transmitarray overall antenna efficiency. In this work, the hemispherical antennas offer distinct advantages for deployment in wireless lens of the proposed transmitarray antenna introduces an communication systems. additional degree of freedom for wavefront control, thereby They generally provide higher antenna gain, higher an- achieving a wider beam-steering range than conventional

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Conformal FCC Gold Resonators: The FCC elements function as localized resonant FSS elements electromagnetically coupled to the underlying graphene sectors. The scaling factor α was optimized through a parametric sweep to establish an appropriate trade-off between geometric miniaturization and electromagnetic performance. Extremely small scaling factors, such as α = 0.1, were excluded because the pronounced dimensional reduction between successive fractal levels imposes geometric constraints that limit the ability of the fifth-ring elements to maintain the required transmission amplitude and phase responses. In contrast, α = 1 was also excluded since the dimensional scaling between adjacent rings is essential for achieving the desired bandwidth enhancement and fully exploiting the advantages of the fractal geometry. Accordingly, Fig. 1: The hemispherical transmitarray antenna comprises 121 the range 0.3 ≤ α ≤ 0.9 was selected for the parametric FCC elements arranged across the curved aperture, including optimization. A systematic parametric sweep of the candidate a central apex element surrounded by five concentric rings scaling factors showed that α = 0.8 provides the most favorable containing 8, 16, 24, 32, and 40 patches, respectively. A scaling balance between fractal miniaturization and electromagnetic factor of 0.8 is applied between successive rings to gradually performance for a = 8. The resulting geometry and ringscale the patch dimensions across the aperture. The proposed dependent dimensions are summarized in Table I of [40]. stacked multilayer transmitarray antenna is composed of a Further design principles and electromagnetic advantages of SiO2 substrate with a thickness of 10 × 10−6 m and εr,SiO2 = the FCC elements, particularly their contribution to bandwidth 3.9, followed by an hBN dielectric layer with a thickness of enhancement and resonant response engineering, are discussed 0.25 × 10−6 m, graphene sectors, another hBN dielectric layer in detail in [40], [41]. Graphene Sectors: The graphene sectors provide electrically with a thickness of 0.25 × 10−6 m and εhBN = 4.4, and a top tunable surface impedance via voltage-controlled modulation layer of FCC patches fabricated from gold (Au) and exposed of the chemical potential. They function as voltage-controlled to free space. The FCC patch geometry and its fractal topology tunable impedance surfaces, where the applied bias voltages are illustrated in the inset for further clarification [40], [41]. independently regulate the chemical potentials of the graphene remainder of this paper is organized as follows. Section II sectors, thereby modifying their complex surface conductivities presents the design objectives and physical architecture of the and enabling dynamic control of the electromagnetic response proposed hemispherical graphene-based transmitarray antenna. of each unit cell. This electrical tunability provides dynamic Section III is devoted to the CST Studio Suite visual basic for control over local transmission amplitude and phase responses, applications (VBA) macro developed for automated modeling enabling programmable beam steering. hBN Dielectric Layers: The proposed stacked multilayer and construction of the multilayer hemispherical transmitarray. Section IV provides the performance evaluation and discussion, architecture employs two hBN dielectric layers, designated as including beam-steering characteristics, radiation performance, hBN1 and hBN2 . The hBN1 layer is positioned between the antenna efficiency, and impedance bandwidth. Finally, Section SiO2 substrate and the graphene sectors, serving as an electrical isolation layer and dielectric spacer. During the fabrication V presents a brief conclusion of this work. process, microchannels can be implemented within hBN1 to II. D ESIGN S PECIFICATIONS AND P HYSICAL provide independent voltage biasing of individual graphene A RCHITECTURE sectors without altering the SiO2 supporting layer. The hBN2 Our proposed antenna is a feedhorn-excited programmable layer, positioned between the graphene sectors and FCC gold hemispherical transmissive metasurface lens operating at 250 resonators, prevents direct electrical contact and short circuits GHz, Fig. 1. The lens modifies the aperture field distribution while preserving the required electromagnetic coupling for to achieve an enhancement in its radiation pattern control, tunable unit-cell operation. a reduction in its sidelobe levels (SLLs) by suppressing SiO2 Supporting Layer: The hemispherical dielectric subthe radiation pattern away from the desired main beam and strate of SiO2 provides mechanical support for the conformal an increased angular discrimination between the main lobe metasurface. Moreover, this hemispherical profile introduces and sidelobes. Our proposed hemispherical lens is realized controlled dielectric loading and enforces the physical curvature as a conformal stacked multilayer architecture consisting necessary for efficient radial-to-planar wavefront transformaof a SiO2 dielectric supporting substrate, voltage-controlled tion. graphene sectors, ultra-thin hexagonal boron nitride (hBN) Circular Feedhorn Antenna: The excitation feedhorn, with dielectric isolation layers, and fractal concentric circular (FCC) an aperture diameter of 5 mm, is positioned at the geometric gold resonators [40], serving as frequency-selective surface center of the hemispherical aperture so that the incident elec(FSS) elements, as illustrated in Fig. 1. The electromagnetic tromagnetic wave illuminates the conformal metasurface prefunctionalities and contributions of the individual constituent dominantly along the local radial direction. This geometrically layers are outlined as follows: symmetric placement minimizes aperture phase nonuniformity, planar reflectarrays, Table I, and transmitarrays, Table II. The

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equalizes the propagation path lengths to the FCC resonant patches, and maximizes the electromagnetic coupling efficiency between the feed source and the hemispherical transmitarray, thereby improving wavefront uniformity and transmission performance. Unlike conventional RISs, our proposed antenna, Fig. 1, operates as a transmissive programmable metasurface lens. As the electromagnetic wave radiated by the feedhorn propagates through the stacked multilayer hemispherical aperture, it interacts with the electrically tunable graphene sectors and FCC resonators to modulate the output field. By dynamically adjusting the local transmission profile of individual unit cells, the transmitted wavefront is reshaped to synthesize the desired far-field radiation pattern with electronically reconfigurable beam-steering capabilities.

hBN dielectric layer, the transmitted phase can be precisely controlled beyond the graphene conductivity variation alone. This additional reactive tuning mechanism provides an extra degree of freedom for phase engineering and enables the realization of a smooth phase gradient over the metasurface aperture. It is worth emphasizing that the coupling between the resonant modes of the hBN layer and the intraband and interband electronic transitions in graphene becomes significant primarily at higher frequencies, particularly in the infrared and optical regimes, where the photon energy is sufficient to excite these electronic transitions. However, at 250 GHz, the photon energy remains below the interband transition threshold, and the graphene response is predominantly governed by the tunable intraband Drude conductivity. Hence, graphene conductivity is dominated by the intraband Drude contribution, while the interband effect is negligible. Because hBN dielectric excitation occurs only in the infrared and optical regime, the hBN layer III. A NALYTICAL F RAMEWORK FOR THE cannot resonantly couple to the graphene layer and therefore VOLTAGE -C ONTROLLED S URFACE I MPEDANCE AND does not provide an additional degree of freedom for the phase E QUIVALENT RLC C IRCUIT OF S TACKED M ULTILAYER tuning at 250 GHz. Consequently, the response of the stacked U NIT C ELLS multilayer unit cells is governed primarily by graphene, where The electromagnetic conductivity of graphene is described by modulating the graphene chemical potential alters its intraband the Kubo formalism, which consists of intraband and interband conductivity to enable dynamic transmission phase control. contributions [42], [43]. The intraband conductivity originates Because the hBN layer behaves as a conventional dielectric from the Drude-like response of graphene charge carriers rather than a resonant medium at 250 GHz, it does not introduce and governs the electromagnetic behavior of graphene at additional resonant phase modulation into the response of microwave and terahertz frequencies. This contribution enables the stacked multilayer unit cells. Consequently, beam-steering electrically tunable conductivity through modulation of the and beam-squint mitigation mechanisms associated with hBN graphene chemical potential, thereby providing dynamic control dielectric resonances, which become significant only at higher over the surface impedance and electromagnetic response. frequencies, are limited in the present sub-terahertz or lowIn contrast, the interband conductivity is associated with terahertz regime and cannot be effectively utilized. At 250 GHz, electronic transitions between the valence and conduction the tunable electromagnetic response is therefore dominated by bands and primarily contributes at higher frequencies, such as the voltage-controlled graphene conductivity and its associated the infrared and optical regimes, where interband excitation surface impedance variation. becomes dominant. In the following, we briefly review the analytical framework Therefore, the intraband term alone accurately approximates of our transmitarray antenna. The comprehensive theoretigraphene conductivity in most terahertz transmitarray applica- cal formulations, their implications, and the corresponding tions, while the interband contribution is negligible for most formulation derivations are discussed in detail in [41]. The terahertz transmitarray applications. beam-steering capability of the proposed transmitarray antenna, For illustrative purposes, if the operating frequency were Fig. 2, originates from the electrically tunable graphene shifted from 250 GHz to the infrared or optical regimes, surface impedance. The graphene surface impedance can be dielectric excitation of the hBN layers would become significant. dynamically controlled by the intraband Kubo model, where In this sense, inserting an hBN layer between FCC metal the chemical potential µc modulates the static conductivity of patches and graphene introduces resonator regions, providing σ0 (µc ) to determine the overall complex conductivity of the additional phase variations across the hemispherical curvature graphene sheet [42], [40], [41], of In other words, varying the bias voltages applied to the e2 kB T graphene sectors modifies not only does the graphene surface σintra (ω, µc , Γ, T ) = −j 2 conductivity but also the electromagnetic coupling with the πℏ (ω − j2Γ)    surrounding multilayer structure. µc −µc /(kB T ) × + 2 ln 1 + e In other words, not only does a variation in the bias voltages kB T applied to the graphene sectors modify the graphene surface (1) conductivity but it also modifies the electromagnetic coupling where µc , τ , T , kB , ℏ represent the graphene chemical potential, with the surrounding multilayer structure. In particular, the carrier relaxation time, temperature, Boltzmann constant, and resonant response of the hBN dielectric layer between the reduced Planck constant, respectively. Γ is related to τ by 1 metallic FCC patches and graphene introduces additional Γ = 2τ . According to the intraband (Drude-like) Kubo reactive loading, providing an extra degree of freedom for conductivity of graphene in Eq.1, the graphene conductivity controlling the transmission phase. By electrically tuning the varies continuously with the chemical potential, which is resonant interaction between the graphene sectors and the controlled by the voltages applied to the graphene sectors.

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The electrostatic relation between the voltage and graphene chemical potential is expressed as [41], r ℏvF πCg µc,(n,k) (eV) = Vg,(n,k) − VD (2) e e where vF , Cg , Vg,(n,k) , and VD denote the Fermi velocity of graphene, the gate capacitance per unit area, the applied gate voltage of the (n, k)-th unit cell, and the Dirac-point voltage, respectively. At 250 GHz, considering only the graphene sheet without the surrounding multilayer structure, the equivalent surface impedance is directly determined from the surface conductivity using the intraband Kubo formulation, as follows [41], Zs (ω, µc , Γ, T ) ≈

1 σintra (ω, µc , Γ, T )

(3)

The graphene surface conductivity is governed primarily by the intraband Drude response, while the interband contribution is negligible over the considered sub-THz range. In this case, the resistive component originates from finite carrier scattering, whereas the imaginary component represents the kinetic inductance associated with the inertia of charge carriers. Since the lateral dimensions and total thickness of each stacked multilayer unit cell are electrically small compared with the operating wavelength, the multilayer structure including FCC patch layer, hBN dielectric isolation layers, and supporting substrate can be accurately modeled as an equivalent impedance sheet characterized by the following complex surface impedance [41],
Dξ′ 1 + ω 2 τ 2 τ Dξ ′ Zs,(n,k),ξ (Vg ) = +jω Gξ σ0,(n,k) Gξ σ0,(n,k) {z } | | {z } Rξ (Vg )

j − ω

Lξ (Vg )

1 Cgeo,(n,k),ξ |

+

πℏ2 vf2

!

2e2 Dθ Dϕ µc,(n,k) {z } 1 Ceff,ξ (Vg ) (4)

where ξ ∈ {θ, ϕ} represents the polarization index, corresponding to the θ- or ϕ-polarized electromagnetic field component and ξ ′ denotes the orthogonal periodicity (e.g., for Rθ , Dξ′ = Dϕ ). Here, Dθ and Dϕ denote the angular discretization factors associated with the dimensions of sectors along the curved aperture. The effective capacitance of Ceff,ξ consists of two components of (a) geometrical capacitance Cgeo,(n,k),ξ and (b) quantum capacitance Cq . The conductivity variation changes the graphene resistance Rξ , kinetic inductance Lξ , and quantum capacitance Cq . Independently tuning the impedance of spatially distributed graphene sectors synthesizes the phase profile required to steer transmitted electromagnetic waves toward directions of moving targets. The multilayer stack surrounding the graphene sectors modifies the local electromagnetic field distribution and current flow, introducing additional equivalent inductive and capacitive loading that influences the resonant patches and the resulting effective surface impedance. Hence, the

Fig. 2: The hemispherical transmitarray comprises 121 independently tunable graphene sectors distributed across the curved aperture, where each unit cell is modeled as a locally planar trapezoidal sector. Each graphene sector is independently biased through a gate voltage to dynamically regulate its chemical potential, enabling tunable electromagnetic responses over the range of 0.1 eV ≤ µc,(n,k) ≤ 1 eV. All unit cells are characterized by a common carrier relaxation time of τ = 0.1ps and an operating temperature of T = 300K. The graphene sectors are physically and electrically isolated from neighboring elements, ensuring independent electrostatic bias control and reliable reconfigurable electromagnetic operation [41].

complete voltage-controlled impedance model of the stacked multilayer transmitarray unit cell is established by incorporating the electromagnetic loading effects of the graphene sectors and surrounding dielectric and metallic layers, consistent with Fig. 1, in the following equation [41], Zcell (Vg ) = Zg (Vg ) + ZhBN2 + ZSiO2 +

1 jωCg

(5)

Consequently, the complete electromagnetic response of the unit cell can be efficiently represented by an equivalent RLC circuit model in Fig. 3. Accordingly, the impedance contributions of

Fig. 3: The equivalent circuit model of our proposed hemispherical transmitarray antenna corresponding to Fig. 1. The model includes the dielectric capacitances of the SiO2 substrate, CSiO2 , and upper hBN layer, ChBN2 , while the graphene sheet impedance is represented by a series Rg -Lg branch with an additional gap capacitance Cg . The lower hBN layer is omitted because it contributes only a fixed capacitive loading effect due to its invariant dielectric properties [41] . the SiO2 substrate and upper hBN dielectric layer are given by [41],
η0 √ ZSiO2 = j √ tan k0 εr,SiO2 tSiO2 (6) εr,SiO2

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ZhBN2 = j √

η0 √ tan k0 εr,hBN thBN2 εr,hBN

(7)

characteristics. The angular steering errors relative to the desired beam direction of (Θ0 , Φ0 ) are defined as,

where η0 and k0 refer to the free-space impedance and the freeδΘ = Θ(pk) − Θ0 (12) space wavenumber, respectively. t(SiO2 ) and t(hBN2 ) denote δΦ = min Φ(pk) − Φ0 + 2πm (13) the thicknesses of the corresponding dielectric layers. This m electrically controlled conductivity modulates the effective surface impedance of each transmitarray unit cell, which subse- where Eq.13 consider the periodicity of the azimuth angles. quently tunes the transmission coefficient, S21 , particularly its m denotes an integer used to account for the periodicity of the typical angular pointing transmission phase, thereby enabling dynamic beam steering the azimuth angle. In this work, ◦ errors remain within δΘ ≤ 1 for elevation steering beam and of the transmitted wavefront [22], [21], [12], in the following ◦ ◦ 3 ≤ δΦ ≤ 5 for azimuth steering beam. sequence of the tuning mechanism [41], The relative gain degradation, Grel , is another important Vg → µc → σg → Zs → ∠S21 → Beam Steering (8) metric for evaluating beam steering performance. It quantifies the scan loss by comparing the maximum magnitude of the array factor at the steered direction with the broadside reference A. Spherical Array Factor and Beam Performance Metrics response, as expressed by, In contrast to planar arrays, where the phase distribution can ! |AF (Θ(pk) , Φ(pk) )| be described by a linear phase gradient along two orthogonal Grel = 20 log10 (14) ref axes, our proposed hemispherical aperture requires a 3D phase |AFbroadside | formulation due to the nonplanar distribution of the radiating elements. Each unit cell introduces a unique phase contribution where AF (Θpk , Φpk ) represents the maximum magnitude of actual peak radiation direction after determined by its spherical position and propagation path. the array factor at the ref beam steering. AF broadside denotes the reference peak of Therefore, the conformal array factor should incorporate the the magnitude of the array factor at the broadside direction, exact 3D coordinates of all elements to accurately predict the ◦ ◦ typically (Θ, Φ) = (0 , 0 ), which is used as the reference far-field radiation characteristics. Accordingly, the conformal level for quantifying scan loss. The relative gain degradation array factor is expressed as [41], metric directly quantifies a reduction in the peak magnitude of Nr X Nn X the array factor resulting from the beam scanning performance Wn,k ejk0 ŝ(Θ,Φ)·r(n,k) AF (Θ, Φ; p) = W0 ejk0 ŝ·r0 + relative to the broadside reference response. Specifically, | {z } n=1 k=1 Apex element Grel = 0 dB represents the ideal scanning performance with no | {z } Concentric ring elements degradation from the broadside response, whereas Grel = −3 (9) dB corresponds to the 3 dB scanning loss, indicating that where ŝ(Θ, Φ) and r(n,k) represent the observation unit vector the steered beam retains 50% of the broadside power level. and the position vector of the (n, k)-th unit cell on the Moreover, values approaching G = −5 dB indicate more rel hemispherical aperture, respectively. The complex excitation severe degradation but may remain acceptable for applications coefficient of each element is expressed in the following demanding extremely wide-angle beam coverage. Generally, equation, [41], this metric is widely regarded as a practical criterion for acceptable gain degradation resulting from beam scanning W(n,k) = ∆Sn Fhorn (θn , ϕ(n,k) )Fcell,(n,k) (Θ, Φ)S21,(n,k) (10) in phased-array and metasurface antenna systems. For practical where ∆Sn , Fhorn (θn , ϕ(n,k) ), and Fcell,(n,k) (Θ, Φ) denote the beam-steering applications, Grel values within the range of effective area of the (n, k)-th unit cell, the incident field distri- −5 dB ≤ Grel ≤ −3 dB are typically regarded as acceptable bution illuminated by the feed horn, and the radiation pattern of levels of the gain degradation resulting from beam steering the corresponding unit cell, respectively. S21,(n,k) incorporates performance. The SLL quantifies the relative suppression of the undesired the programmable transmission amplitude and phase of the graphene unit cell. During the initial optimization stage, the radiation with respect to the peak main-lobe response and element pattern may be approximated by Fcell,(n,k) ≈ 1, to serves as an important metric for evaluating beamforming focus primarily on the aperture illumination and programmable quality, interference mitigation, and spatial selectivity.   transmission response. After incorporating the effects of max |AF (Θ, Φ)| (Θ,Φ)∈Ωsidelobe  feedhorn illumination, the conformal hemispherical geometry, SLL = 20 log10  (15) |AF (Θ(pk) , Φ(pk) )| and the graphene transmission characteristics, the actual mainbeam direction is determined by the direction of maximum radiated power and is expressed as,

where Ωsidelobe represents the angular region excluding the main beam. For practical beam-steering applications, SLLs (Θ(pk) , Φ(pk) ) = arg max |AF (Θ, Φ)|2 (11) in the range of approximately −8 to −10 dB are considered Θ,Φ acceptable for the effective sidelobe suppression. Consequently, However, the beam-steering performance cannot be fully a steering direction is regarded as successfully synthesized characterized by the main-beam direction alone. A compre- when it simultaneously satisfies the four aforementioned criteria. hensive assessment requires additional beam-quality metrics These metrics quantify not only whether the desired beam that evaluate steering accuracy, gain degradation, and sidelobe direction is achieved but also the quality of the synthesized

8

radiation pattern in terms of pointing accuracy, scan efficiency, and sidelobe suppression. B. Antenna Efficiency The lens transmission efficiency is defined as the average transmission efficiency of the stacked multilayer hemispherical antenna under horn illumination and is expressed as, X ∆S(n,k) |Fhorn,(n,k) |2 |S21,(n,k) |2

Studio Suite. For the first time in this work, we developed macro to automatically construct the geometry of the stacked multilayer lens, graphene sectors, material assignments, and simulation environment, eliminating repetitive manual modeling and facilitate efficient parametric design and optimization. A. Macro-based Definition of the Lens Geometry

Following the validation of the metasurface antenna, without the multilayer stack, illuminated by the circular feedhorn in (n,k) X ηlens ≈ (16) the CST Studio Suite environment, the custom VBA macro is developed to automate the construction of our proposed ∆S(n,k) |Fhorn,(n,k) |2 multilayer transmitarray antenna. (n,k) To automate the model generation process, the macro codes where ∆S(n,k) denotes the surface area of the (n, k)-th unit first acquires the geometric, constitutive, and discretization cell. The weighting term accounts for the non-uniform horn parameters through user-defined InputBox dialogs, Table III. illumination across the curved aperture. Furthermore, the Using these inputs, the macro code automatically generates overall radiation efficiency, which accounts for the total power the required CST components, assigns the corresponding conversion efficiency of the antenna system, is expressed as, material properties, and constructs the complete multilayered GdBi −DdBi G transmitarray antenna, including the graphene sectors and other 10 ηrad = = 10 (17) layers in Fig. 1. The macros are developed in VBA and executed D where G and D denote the antenna gain and directivity, directly within CST Studio Suite to automate electromagnetic respectively. This metric quantifies the fraction of accepted modeling and simulation tasks. Our CST VBA macro script power effectively radiated by the antenna after considering all automatically constructs the complete 3D electromagnetic dissipative mechanisms, including conductor losses, dielectric model by generating the geometry, assigning the material losses, and surface-wave losses within the multilayer structure. properties, defining the electromagnetic boundary conditions, It can be extracted directly from CST Studio Suite or computed and configuring the ports and excitation settings. Beyond the automated generation of the electromagnetic using simulated gain and directivity. model, the developed macro codes provide a complete simOn the other hand, the mismatch efficiency is defined as, ulation workflow by configuring and controlling the mesh ηmismatch = 1 − |S11 |2 (18) generation process, executing full-wave electromagnetic simuwhich represents the fraction of incident power delivered to lations, and automatically extracting and exporting the required the horn input after accounting for power reflected due to simulation results. This automated implementation eliminates repetitive manual impedance mismatch. Consequently, the total antenna efficiency modeling, ensures consistent and reproducible geometry generis expressed as, ation, and significantly reduces model setup time. Moreover, ηtotal = ηmismatch ηrad (19) the macro enables efficient parametric studies, optimization, and large-scale design sweeps by automatically regenerating It is worth noting that Eq.19 holds only when ηrad excludes the entire model for different sets of design parameters while the mismatch efficiency contribution. In CST Studio Suite, the maintaining a consistent modeling procedure. reported realized gain and total efficiency inherently account Prior to initiating the geometry generation process, the for the impedance mismatch at the input port. Consequently, macro executes an input validation routine to ensure the the mismatch efficiency is already included in these quantities correctness and physical feasibility of the defined parameters. and should not be applied again in subsequent calculations. Specifically, it verifies that the hemispherical radius, dielectric Alternatively, when the realized gain is available, the total shell thicknesses, and relative permittivity values are positive, efficiency can be obtained as, while ensuring that the combined thickness of the inner Grealized,dBi −DdBi Grealized 10 ηtotal = = 10 (20) dielectric shells does not exceed the hemispherical radius. D Collectively, these efficiency metrics enable a comprehensive B. Dielectric Shells and Layer Interpretation characterization of the power-transfer and radiation perforThe macro automatically generates three concentric, paramemance of our proposed feedhorn hemispherical graphene-based terized hemispherical dielectric shell regions and assigns them transmitarray antenna, Table VIII. to the CST components. Each shell is constructed through a Boolean subtraction process. Two concentric spheres are IV. CST VBA M ACRO FOR AUTOMATED M ULTILAYER initially created, and the inner sphere is subtracted from the H EMISPHERICAL T RANSMITARRAY M ODELING outer sphere to form a spherical shell. A vacuum brick operation To facilitate efficient and reproducible electromagnetic mod- is subsequently applied to remove the lower half, resulting in eling of the proposed multilayer hemispherical transmitarray a true hemispherical dielectric shell. This approach generates antenna, a dedicated VBA macro was developed in CST a true hemispherical dielectric shell geometry rather than a

9

TABLE III: Definition of design parameters used by the CST macro Design parameter Hemisphere radius, R Cell distribution Inter-cell gap Apex cell radius Gold-patch radial offset

Inner-shell thickness of dis2 Additional inner-shell thickness of dis0 Additional inner-shell permittivity Inner-shell permittivity Outer-shell permittivity

Corresponding CST model parameter definition Radius of the hemispherical lens Number of graphene sectors distributed along each ring Isolation spacing between adjacent graphene sectors Radius of the apex graphene sector Radial offset extending from R to R + dis and also defines the radial placement offset of the conformal gold patches Thickness of the inner dielectric shell spanning from R − dis2 to R Radial thickness of the additional inner dielectric shell, R − dis2 − dis0 ≤ r ≤ R − dis2 Relative permittivity of the additional inner dielectric shell Relative permittivity of the inner dielectric shell Relative permittivity of the outer dielectric shell

complete spherical layer, providing an accurate representation of the conformal lens structure. It is worth mentioning that the current macro employs generic isotropic dielectric materials for the generated shell regions. If the physical multilayer stack is intended to explicitly represent hBN and SiO2 , the existing shell-generation methodology can be retained without modification. However, these placeholder material definitions should be replaced with the corresponding physical material properties, including the appropriate relative permittivity, loss tangent, and, when required for accurate hBN modeling, anisotropic dielectric tensor parameters.

where Nn and Nr represent the number of rings, excluding the apex unit cell, and the number of unit cells distributed on each ring, respectively. The azimuthal shrinkage is scaled by the local spherical metric coefficient, sin(θC ), where θC represents the central polar angle of the cell. This correction ensures a uniform physical isolation gap across the curved hemispherical surface by compensating for the variation in azimuthal arc length of Lϕ = R sin(θ)∆ϕ. In the vicinity of the hemispherical apex, sin(θ) approaches zero, potentially causing numerical instability and cell geometry collapse during the discretization process. To avoid this issue, the macro enforces a minimum value for the scaling factor, thereby ensuring robust and accurate generation of apex-adjacent graphene cells. The main features of the graphene-sector generation procedure are summarized as follows: (1) apex graphene cell: the apex graphene cell is generated independently as a circular polygon conformal to the hemispherical surface. It is discretized using 96 segments to provide a smooth and accurate approximation of the curved boundary while maintaining geometric consistency with the surrounding graphene cells, (2) ring graphene sectors: graphene sectors are generated as four-vertex Polygon3D curves in the spherical coordinate and subsequently converted into planar sheet faces using the CST CoverCurve operation, (3) electrical isolation: gaps are introduced between adjacent graphene sectors to electrically isolate the individual graphene regions, thereby enabling independent electrostatic biasing and programmable control of the local surface impedance, and (4) independent material assignment to each graphene sector: each graphene sector is assigned a dedicated material definition and an independent tuning parameter of V(n,k) , thereby enabling independent control of its chemical potential and corresponding electromagnetic response. This macro-based generation method enables accurate conformal discretizations of the hemispherical graphene metasurface while preserving the independent tunability required for programmable beam steering and wavefront control.

C. Graphene Sector Generation on the Hemispherical Surface The macro script automatically instantiates the graphene D. Graphene Material Definition and Chemical Potential sectors within the dedicated component folder in the CST Assignment environment. The macro handles the hemispherical discretizaFor each graphene unit cell, the macro assigns independent tion by treating the apex cell separately and then constructing tuning parameters of V(n,k) and generates a corresponding the remaining graphene cells sequentially in a ring-by-ring graphene material definition. The macro indeed evaluates the distribution over the curved surface. This approach ensures Kubo conductivity internally over a wide frequency range accurate conformal placement of the graphene elements while and converts the complex surface conductivity into tabulated maintaining independent electrical control of each unit cell. surface impedance values. Graphene material properties are This macro-driven layout procedure ensures high-fidelity implemented automatically in CST through a macro subconformal placement across the hemispherical surface while routine that executes four primary tasks: (1) conductivity preserving localized, independent electrostatic bias control for evaluation: it computes graphene surface conductivity over a each graphene unit cell. To introduce an electrical isolation gap broad frequency range using the Kubo model, (2) impedance between adjacent graphene sectors, the macro slightly shrinks conversion: it transforms the complex surface conductivity the angular boundaries before constructing each cell according into equivalent surface impedance values that are tabulated for to, the CST environment, (3) material creation: it instantiates the ∆θ ∆θ iso θniso = θn + , θn+1 = θ(n+1) − (21) corresponding material definitions within the CST environment, 2 2 and (4) property assignment: it automatically instantiates the CST material definitions and maps the frequency-dependent ∆ϕ ∆ϕ ϕiso , ϕiso impedance data directly to the tabulated material properties. (n,k) = ϕ(n,k) + (n+1,k+1) = ϕ(n+1,k+1) − 2 2 (22) This formulation provides independent tuning of individual

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TABLE IV: Variable quantities within macro subroutines and the macro execution routine Variable Quantities within Macro Subroutines Voltage-control parameters: V(1,1) , . . . , V(Nr ,Nn ) Graphene sector material definition Thickness of graphene sector

each

Temperature of each graphene sector Carrier relaxation time

Chemical potential of each graphene sector

Macro Execution Routine

An independent parameter V(n,k) is assigned to each graphene sector material and directly defines the sector-specific chemical potential used in the Kubo conductivity evaluation Equivalent lossy-metal material definition with tabulated transparent surface impedance values Effective graphene thickness parameter used for equivalent permittivity conversion The temperature parameter incorporated into the Kubo formulation to evaluate graphene surface conductivity Carrier relaxation-time parameter τ used to model scattering losses in the graphene Kubo conductivity formulation The chemical potential parameter is directly scaled by V(n,k)

graphene sectors, allowing the hemispherical metasurface to realize spatially inhomogeneous impedance distributions for dynamically programmable wavefront control. The main macro parameters associated with the graphene material definition are summarized in Table IV. Although the macro labels voltage variables as V(1,1) , . . . , V(Nr ,Nn ) , our implementation directly interprets these parameters as the graphene chemical potentials expressed in electron volts. Consequently, V(n,k) represents the chemical potential of the i-th graphene sector rather than an applied voltage V(n,k) ≡ µc,(n,k) . A voltage-controlled interpretation would require an additional electrostatic model to establish the relationship between the gate bias Vg and the induced chemical potential µc (Vg ). If the final physical model is intended to describe voltage-controlled graphene tuning, the macro should either rename these parameters as µc,(n,k) or incorporate the electrostatic relationship between gate voltage and chemical potential using the gate capacitance of the dielectric layer (e.g., hBN-based gating structure). The intraband term dominates at our operating frequency of 250 GHz and typical chemical potentials of 0.2–0.8 eV, the implemented full Kubo-type numerical evaluation ensures accuracy and validity across a broader terahertz (THz) spectrum.

(2) Algorithmic flow: the macro generates the gold-patch distribution by creating one copy of the reference patch for each graphene sector. Each copied patch subsequently undergoes a sequence of geometric transformations to achieve conformal placement on the hemispherical surface: duplication: a copy of the reference patch is created, scaling: the copied patch is resized according to the prescribed scaling factor, orientation: the patch is rotated about the local coordinate axes to align with the target graphene sector, where the rotation angles are determined by the corresponding spherical coordinates of (θ, ϕ), and positioning: the transformed patch is translated to the cartesian coordinate system corresponding to the center of the target graphene sector. (3) Spatial mapping: the spatial placement of the gold patches is determined according to the hemispherical cell distribution: apex cell: the gold patch associated with the apex graphene cell is positioned at the hemispherical apex using the default spherical coordinates (θ, ϕ) = (0, 0), ring cells: for all non-apex graphene sectors, each gold patch is positioned at the center of the corresponding graphene sector using its spherical coordinates of (θC , ϕC ), and radial placement: the patch center is evaluated at a radial distance r = R+dis, ensuring that the gold patch is located above the graphene layer and dielectric spacer while preserving the intended multilayer configuration, and scaling law and parametric control: the default implementation employs geometric scaling across successive rings to the patch dimensions, patchScale = 0.8ringIndex , resulting in a gradual reduction in patch size with increasing ring index to account for the spatially varying local geometry of the hemispherical surface. The graphene conductivity is initially calculated, and the corresponding tabulated surface-impedance and equivalentpermittivity materials are created during the generation of the graphene sectors. After the resonator patches are added to the previous configuration, the existing tabulated surfaceimpedance and equivalent-permittivity materials are updated to account for the modified unit cells. F. Macro Architecture and Subroutine Functionality The CST VBA macro features a modular architecture comprised of dedicated subroutines, each executing a distinct phase of the hemispherical metasurface construction. This design enhances code readability, simplifies maintenance, facilitates future feature extensions, and allows independent validation of geometry-generation and material-definition routines. The core subroutines and their specific operational roles are summarized in Table V.

E. Gold Patch Placement on the Hemispherical Surface The macro employs a predefined reference metallic patch as a template for generating the corresponding elements across the aperture. For each unit cell, the reference geometry is replicated G. Horn and Hemisphere Alignment After generating the hemispherical lens geometry, the horn and subsequently scaled, rotated, and translated according to the corresponding cell dimensions and spatial coordinates, thereby antenna is positioned such that its phase center is located at, or very close to, the origin of the hemispherical coordinate system. constructing the successive active rows. (1) Prerequisite: prior to executing the macro, a reference This center-fed configuration ensures uniform illumination gold patch should be defined within the active CST project. because the distance from the horn’s phase center to the This reference patch serves as the template for generating hemispherical aperture remains approximately constant. Conthe conformal gold-patch array and should exist in the model sequently, the incident spherical wavefront maintains a nearly before the macro is executed. uniform phase distribution across the lens surface, minimizing

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TABLE V: Generalized functions of the CST VBA macro subroutines Subroutine Main

Input parser

Material definition

Dielectric shell generation Coordinate transformation Apex cell generation Graphene cell generation Gold patch placement

Graphene material modeling

Functionality Collects user inputs, validates design parameters, generates the antenna geometry, assigns material properties, and constructs the conformal multilayer model. Converts user-defined cell distributions into numerical arrays specifying the number of unit cells in each hemispherical ring. Creates dielectric and graphene material models using the specified constitutive parameters. Constructs concentric hemispherical dielectric shells from user-defined geometric parameters. Transforms spherical coordinates into Cartesian coordinates for geometry generation and object placement. Constructs the graphene unit cell located at the hemispherical apex. Generates conformal graphene unit cells on the hemispherical surface according to the prescribed discretization. Replicates, scales, rotates, and positions the reference gold patch at the center of each graphene sector. Evaluates the graphene conductivity using the Kubo model and generates the corresponding frequency-dependent equivalent material properties.

corresponding graphene unit cell, ψtrans,(n,k) = ∠S21,(n,k) , yielding, ψtotal,(n,k) = ψincident,(n,k) + ψtrans,(n,k)

(25)

Therefore, the transmitted wavefront no longer matches the desired phase distribution, causing beam-pointing errors, degraded beam-steering performance, elevated sidelobe levels, and a reduction in the achievable antenna gain. To overcome this limitation, and thereby achieve constructive interference along the desired beam-steering direction, the transmitted phase profile should incorporate the actual incident wavefront distribution, i.e., Eq.24, and satisfy the near-field phase-map condition, ψincident,(n,k) + ψtrans,(n,k) ≈ ψreq,(n,k)

(26)

This phase distribution defines the required transmission phase response of the programmable graphene-based unit cells to achieve the desired beam-steering functionality. For CST verification of the near-field phase map over the hemispherical coordinates, first the minimum horn-to-lens distance is estimated using the far-field (Fraunhofer) criterion, d≥

2 2Dhorn λ

(27)

where Dhorn denotes the maximum aperture dimension of the feed horn antenna. The hemispherical lens is then positioned at the calculated minimum horn-to-lens distance within the CST Studio Suite environment. Subsequently, the phase and the phase compensation required by the metasurface elements. amplitude of the incident electric field are evaluated at the To ensure uniform and symmetric illumination of the aperture, physical lens position as follows, the horn boresight should be aligned with the +z-axis when the hemispherical lens aperture is oriented toward the positive (28) E(θ, ϕ) = |E(θ, ϕ)|ejψinc (θ,ϕ) z-direction. In the CST assembly model, the hemispherical lens should be positioned within the far-field region of the horn where |E(θ, ϕ)| denotes the incident field amplitude and antenna. The far-field criterion establishes the initial horn-to- ψincident = ψinc (θ, ϕ) = ∠E(θ, ϕ) represents the correspondlens separation, while the near-field phase map verifies that ing incident field phase distribution over the hemispherical the electromagnetic field illuminating the hemispherical lens aperture. The near-field verification in the CST Studio Suite maintains sufficient phase and amplitude uniformity. Indeed, environment can be fulfilled by the two criteria of phase under the ideal assumption of a uniform incident phase, the curvature and amplitude uniformity. The amplitude uniformity required transmission phase for the (n, k)-th unit cell is given is evaluated by the spatial variation of the field amplitude by, across the aperture given by, ψreq,(n,k) = −k0 ŝ0 · r(n,k) + C

(23) ∆A =

where k0 , ŝ0 , and r(n,k) denote the free-space wavenumber, the desired beam-steering direction, and the position vector of the (n, k)-th unit cell, respectively. The constant of C represents an arbitrary reference phase. In practice, however, the incident field radiated by the horn antenna possesses a spatial variation in its phase distribution, ψincident,(n,k) = ∠Einc (r(n,k) )

(24)

where Einc (r(n,k) ) denotes the complex incident electric field at the location of the (n, k)-th unit cell. Consequently, the total transmitted phase includes the phase introduced by the

max |E(θ, ϕ)| − min |E(θ, ϕ)| max |E(θ, ϕ)|

(29)

The phase curvature measures the total phase variation introduced by the horn wavefront across the lens aperture, ∆ψincident = max(ψinc (θ, ϕ)) − min(ψinc (θ, ϕ))

(30)

The horn illumination should be verified to ensure consistency with the assumptions of the phase-synthesis formulation. If significant phase curvature or nonuniform amplitude illumination persists across the hemispherical aperture, the horn-to-lens distance should be increased or adjusted until the required illumination conditions are satisfied.

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H. Hybrid Solver Strategy in the CST Environment For our transmitarray antenna, the wavelength propagating inside the SiO2 dielectric layer is given by, λ0 λSiO2 = √ εr

(31)

εr represents the relative permittivity of the SiO2 layer. Hence, the effective wavelength inside SiO2 becomes, λ0 ≈ 0.506λ0 (32) λSiO2 = √ 3.9 For both frequency- and time-domain full-wave solvers in CST Studio Suite, the maximum mesh size should satisfy electromagnetic discretization requirements, and is typically selected to be smaller than λ/5, finer resolutions like λ/10 or λ/20 are used for higher accuracy. Furthermore, in CST Studio Suite frequency-domain simulations, the tetrahedral and hexahedral mesh resolutions are typically set to λ/4, and λ/8, respectively. The maximum allowed tetrahedral mesh edge length under a refined λ/10 resolution criterion is chosen by the following equation, 0.506λ0 λSiO2 = ≈ 0.0126λ0 (33) 4 × 10 40 where ∆mesh,tet denotes the maximum allowed tetrahedral mesh edge length. Therefore, for our transmitarray antenna operating at 250 GHz, the maximum tetrahedral mesh edge length based on this refined criterion is approximately given by, ∆mesh,tet ≤ 15.1 µm (34) ∆mesh,tet ≤

Moreover, nanometer-scale graphene layers, hBN dielectric layers, and a large number of conformal gold resonators can become computationally prohibitive due to the extreme multiscale nature of the problem. Consequently, our simulation constitutes an electrically large electromagnetic problem, particularly within the region between the horn antenna and the lens; conversely, the graphene, hBN, and gold resonator patches are much smaller and locally resonant. Its computational complexity is comparable to the full-wave analysis of largescale platforms, such as maritime vessels with extensive radar cross-sections (RCS). One way to overcome this constraint and substantially reduce computation time is to implement a hybrid solver strategy with a unidirectional assembly of the horn antenna and lens. Our simulation procedure begins by independently modeling and simulating the horn antenna using the time-domain solver. Subsequently, the hemispherical lens, consisting solely of the SiO2 substrate, is modeled and analyzed using the frequencydomain solver. CST Studio solves the coupled model of the horn antenna and the hemispherical lens including only the SiO2 substrate. The computed electromagnetic field distribution is stored and subsequently used as a new excitation source for the subsequent simulation stage. Rather than approximating the excitation as an ideal plane wave, the saved field solution preserves the realistic illumination characteristics of the horn antenna, including its amplitude taper and phase curvature across the hemispherical aperture.

The assembly of these two components can be performed using either a unidirectional coupling approach or a bidirectional coupling approach, depending on the desired level of electromagnetic interaction and computational accuracy. Since the lens is positioned within the far-field region of the horn antenna, multiple-scattering and backscattering effects are considered negligible. Therefore, a unidirectional coupling approach is adopted for our simulation. The remaining transmitarray layers are then added sequentially, and the simulation process is repeated after the incorporation of each layer until the complete multilayer transmitarray antenna is obtained by advancing the simulation in a forward-only manner, without returning to or re-solving the preceding stages. Generally, for an initial engineering workflow, a unidirectional coupling approach provides a practical, computationally efficient first step. This method is particularly suitable when the lens introduces negligible perturbation or detuning to the horn antenna. However, if significant back-reflection, impedance mismatch, or strong near-field interactions are observed, a bidirectional coupling scheme or a full-wave analysis of the entire system should be employed to ensure modeling accuracy, Table VI. I. Electromagnetic Field Preservation and Reuse In the proposed hybrid solver, the horn antenna is first simulated independently; the resulting electromagnetic field distribution at the lens interface is saved and subsequently imported as an equivalent excitation source. The computed field distribution from the horn antenna and lens including only SiO2 , as a new excitation source, can then be imported into a second simulation containing the detailed graphene sectors and gold patches. This approach eliminates the need to repeatedly solve the horn antenna for each graphene voltage sector, significantly reducing the computational cost of voltagesweep simulations. The complex electric and magnetic nearfield distributions should be exported while maintaining a consistent phase reference for subsequent field excitation and coupling analysis. The same coordinate system should be used, reference origin, and lens position when importing the field distribution into the detailed unit-cell model to ensure accurate phase and amplitude alignment. We should verify that the polarization of the imported field is consistent with the local orientation of the metallic patches and the principal direction of the graphene conductivity tensor. Once imported, the electromagnetic field distribution serves as the excitation source for detailed transient or frequency-domain simulation of the graphene–gold patterned lens. Evaluate the resulting far-field radiation patterns across multiple bias voltage matrices to demonstrate the dynamic beam-steering capability of our tunable transmitarray antenna. Although this approach significantly reduces computational cost, the accuracy of the imported field should be verified; any discrepancies in the extracted amplitude or phase distribution directly degrade the synthesized transmitarray aperture field and beam-steering performance. This verification step ensures that the excitation from the imported field accurately reproduces the physical illumination of the original horn antenna prior

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TABLE VI: Hybrid solver of the CST environment for the proposed multilayer transmitarray antenna Stage CST Task 1 Horn antenna and lens including only a smooth hemispherical SiO2 shell 2

3

4

Assembly coupling

Solver A transient solver is used for the horn antenna, while a frequency-domain or asymptotic platform solver models the smooth dielectric shell Configure unidirectional hybrid solver for the horn–lens assembly

Purpose Compute the horn radiation pattern and determine the incident electromagnetic field distribution illuminating the hemispherical lens aperture.

Couple the horn source task with the smooth lens platform task without requiring the simultaneous meshing of all fine transmitarray features. Store the computed near-field distri- Generate a reusable field excitation that serves as a new Electromagnetic field export bution or field-monitor data at the source for the subsequent layers in the transmitarray operating frequency of 250 GHz for antenna. subsequent use as the excitation source Complete multilayer transmitarray A frequency- or time-domain solver Evaluate the impact of the voltage-controlled graphene architecture incorporating graphene driven by field excitations imported sectors and gold patches on the transmission coefficient sectors, gold FCC unit cells, and from prior simulation stages S21 , transmission phase response, and far-field beamhBN interfacial layers steering performance of the transmitarray antenna.

to executing the computationally expensive graphene voltagesweep simulations. Following the successful validation, the imported field executes across the entire graphene-based transmitarray model for the required gate-voltage and chemical-potential sweeps. This ensures that the observed beam-steering behavior is caused by the graphene tunability rather than numerical artifacts from the imported field process. (d)

(a)

(e)

Fig. 5: Variation of S11 magnitudes in terms of frequency for our proposed transmitarray antenna, consistent with Fig. 1, for different beam-steering directions of (a) θ = 0◦ and ϕ = 0◦ , (b) θ = 23◦ and ϕ = 300◦ , (c) θ = 50◦ and ϕ = 150◦ , (d) θ = 78◦ and ϕ = 150◦ , and (e) θ = 77◦ and ϕ = 300◦ To validate the hybrid solver, both configurations should yield nearly identical fields, (b)

(c)

Edirect (θ, ϕ) ≈ Eimported (θ, ϕ)

(35)

AFdirect (θ, ϕ) ≈ AFimported (θ, ϕ)

(36)

The validation procedure compares two equivalent simulation setups: (1) a reference simulation involving direct horn excitation, where the smooth hemispherical SiO2 lens is modeled alongside the original horn antenna. The horn directly illuminates the SiO2 -only lens, and the resulting transmitted nearfield distribution and far-field radiation patterns are recorded, and (2) an imported-field simulation, where the horn antenna is removed, and the same smooth lens is excited using the

14

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4: Demonstration of the wide-angle beam-steering performance of the proposed transmitarray antenna, Fig. 1, through 3D and 2D radiation patterns. The 2D radiation patterns are plotted in the ϕ = 90◦ plane as a function of the elevation angle of θ and the main-lobe magnitude in dBi. The proposed transmitarray antenna provides a full azimuthal scanning range of ϕ = 360◦ and an elevation scanning range from θ = −78◦ to +78◦ at an operating frequency of 250 GHz. The beamsteering directions are illustrated for representative steering angles: (1) for θ = 0◦ and ϕ = 0◦ : (a) 3D radiation pattern with a directivity of 14.72 dBi and (b) 2D radiation pattern with a HPBW of 21.6◦ , (2) for θ = 76◦ and ϕ = 150◦ : (a) 3D radiation pattern with a directivity of 11.70 dBi and (b) 2D radiation pattern with with a HPBW of 40.1◦ , (3) for θ = 78◦ and ϕ = 360◦ : (a) 3D radiation pattern with a directivity of 11.38 dBi and (b) 2D radiation pattern with a HPBW of 41.8◦ , and (4) for θ = 50◦ and ϕ = 150◦ : (a) 3D radiation pattern with a directivity of 13.24 dBi and (b) 2D radiation pattern with a HPBW of 21.3◦ .

field distribution generated by the horn antenna illuminating the hemispherical lens containing only the SiO2 substrate. • Import Nearfield: Import the extracted electromagnetic field distribution as a new excitation source for the remaining graphene–gold transmitarray antenna. • Run Full Sim: Execute the final simulation using the imported source excitation. • Extract Metrics: Record the reflection coefficient S11 and transmission coefficient S21 of the transmitarray antenna. • Analyze Radiation: Run the voltage-matrix configurations and extract the corresponding far-field radiation patterns, beam peak direction, realized gain, SLL, and horn reflection coefficient S11 .

J. CST Simulation Workflow Phase 1: Base Geometry and Macro Execution Initialize Patch: Place Patch17 in Component2 at the coordinate origin. • Verify Properties: Confirm the correct material assignment, thickness, and spatial orientation. • Run Macro: Execute the script to generate the parametric transmitarray layout. • Input Parameters: Specify the hemisphere radius, the number of cells per ring, gap thickness, and top-cell radius. • Define Layer Specs: Enter the exact dielectric thicknesses and material permittivities. •

(g)

(h)

previously saved electromagnetic field distribution obtained from the horn simulation. The corresponding transmitted nearfield and far-field results are then extracted. Good agreement confirms that the imported field preserves the essential characteristics of the horn illumination, including amplitude taper across the hemispherical aperture, incident phase curvature, polarization distribution, and spatial phase reference. • Export Source Field: Save the 250 GHz electromagnetic

Phase 2: Material and Bias Assignment • Audit Structures: Verify the generated Dielectric

Shells, Graphene Cells, and Patch17 shapes. • Assign Dielectrics: Apply specific properties for hBN and SiO2 layers. • Configure Graphene: Set up the chemical potentials of µc for graphene sectors.

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TABLE VII: Radiation performance for different beam-steering angles. These reported antenna efficiencies account for a 5% degradation due to performance imperfections.

accuracy of beam pointing. Moreover, leveraging graphene’s tunable surface impedance and the multilayer electromagnetic interactions embedded in the S21 transmission response extends the available phase-tuning range, enabling larger spatial phase (θ, ϕ) (deg.) Directivity HPBW SLL gradients across the aperture and consequently increasing the (dBi) (deg.) (dB) achievable beam steering angles. (0, 0) 14.72 21.6 −8.5 The results in Table VII demonstrate the capability of our (23, 0) 12.15 39.9 −9.4 (22, 150) 11.96 39.6 −9.6 proposed transmitarray antenna to achieve wide-angle two(23, 300) 12.09 39.4 −9.8 dimensional beam steering while maintaining stable radiation (50, 0) 13.29 22.7 −8.8 characteristics. Some SLL values reported Table VII that drop (50, 150) 13.24 21.3 −7.7 below −8 dB do not imply that the proposed transmitarray (76, 360) 11.38 41.8 −6.8 (78, 150) 11.70 40.1 −9.2 antenna is incapable of tracking moving targets. Nevertheless, (78, 300) 11.43 37.4 −7.7 the corresponding radiation patterns still exhibit a good agreement with the desired beam-steering behavior and can TABLE VIII: Antenna efficiency for different beam-steering support moving-target tracking, but with lower probabilities or directions. reduction in tracking moving targets. The proposed antenna maintains high directivity across θ (deg.) ϕ (deg.) Antenna Efficiency (%) a wide scanning range, including extreme elevation angles. 0 0 79 Specifically, at θ = 78◦ , it achieves directivities of 11.70 23 300 78 50 150 80 dBi for φ = 150◦ and 11.43 dBi for φ = 300◦ . Despite 78 150 71 steering the beam near the horizon, this represents only 77 300 72 an approximately 3-dB reduction from the broadside case. This robust performance demonstrates the effectiveness of the hemispherical transmitarray configuration in preserving • Map Voltages: Implement the graphene bias parameters of V(1,1) , . . . , V(Nr ,Nn ) using the corresponding voltage- aperture efficiency at large scan angles. Although the HPBW increases for large steering angles due to a reduction in effective to-µc conversion model. aperture projection and an increase in phase deviations at Phase 3: Source Setup and Hybrid Assembly extreme scan directions, the proposed antenna preserves stable • Position Horn: Align the feed horn precisely at the center sidelobe performance, with SLL values maintained between of the hemisphere. −6.8 and −9.8 dB. • Check Port: Verify the waveguide port orientation and Table VIII reports the antenna efficiency of the proposed matching. transmitarray antenna for different steering angles of the • Run Primary Sim: Simulate the horn and smooth-platform elevation angle and azimuth angle. The proposed antenna subsystem. maintains high efficiency even at extreme steering directions, • Use Hybrid Solver: Leverage the hybrid assembly solver yielding 71% and 72% at (θ, ϕ) = (78◦ , 150◦ ) and (77◦ , 300◦ ), for the horn antenna and lens including SiO2 . respectively. This corresponds to a reduction of less than Phase 4: Field Export and Final Evaluation 8% compared with the broadside case at (θ, ϕ) = (0◦ , 0◦ ), despite steering the beam toward near-horizon directions. These V. P ERFORMANCE E VALUATION AND D ISCUSSION results confirm the ability of the hemispherical transmitarray The discussion investigates how variations in the applied antenna to maintain efficient power transmission over the voltages across the 121 graphene sectors, Fig. 2, influence the wide scanning range. Furthermore, this high efficiency stems performance of our proposed transmitarray antenna, Fig. 1. It is from combining a low-loss multilayer transmission structure evident from the radiation patterns in Fig. 4 that variations in the with tunable graphene-based unit cells. Phase distortions applied voltages across the 121 graphene sectors substantially induced by scan-angle variations can be significantly mitigated change the main-lobe directions of our proposed transmitarray by leveraging the intrinsic spatial phase compensation of antenna. Compared with the available literature, Tables I and the curved aperture geometry, while the voltage-controlled II, the proposed transmitarray antenna achieves substantially graphene sectors enable precise control of the local transmission greater steering of the main radiation lobes. This behavior phase with minimal amplitude degradation. In addition to the mainly results from the hemispherical lens geometry, which performance degradation caused by imperfections, it is worth introduces additional spatial variations in the S21 transmission emphasizing that the proposed model employs the hybrid solver phase, in conjunction with the bias voltages applied to the and unidirectional coupling setup, under which the effects of graphene sectors. Furthermore, unlike planar reflectarrays, S11 can be effectively mitigated, resulting in an improvement the proposed transmissive configuration benefits from the in the overall antenna efficiency. Consequently, the proposed curved aperture geometry, which provides a broader phase antenna achieves wide-angle beam steering while maintaining compensation capability and enables more effective control of high aperture efficiency, which is critical for tracking moving the outgoing wavefront. Integrating 121 independently tunable targets. graphene sectors enhances phase resolution, allowing finer The impedance matching performance of the proposed adjustment of the local transmission phase and improving the transmitarray antenna for different beam-steering directions is

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illustrated in Fig. 5. The magnitude of S11 at the feedhorn port after integration remains below −10 dB over the entire frequency range from 200 to 300 GHz for all typical steering angles, including broadside and extreme off-axis directions. The superior impedance stability over the wide scanning range is mainly attributed to the FCC gold patches in our proposed antenna. The scaling factor of 0.8 between consecutive FCC rings introduces multiple resonant responses with gradually varying resonant frequencies across the aperture. The overlap of these closely spaced resonances creates a broadband impedance response, allowing the antenna to maintain effective electromagnetic coupling over the entire operating bandwidth. Consequently, variations in the beam-steering directions do not significantly degrade the impedance matching performance. VI. C ONCLUSION In conclusion, we successfully demonstrated the first comprehensive CST Studio Suite implementation of the stacked multilayer hemispherical graphene-based transmitarray antenna operating at 250 GHz. We developed a fully automated CST macro to generate the multilayer hemispherical structure and its 121 independently tunable graphene sectors, establishing an efficient framework for modeling complex, curved, programmable metasurfaces. The numerical results have verified that voltage-controlled graphene sectors enable the wide-angle 3D beam steering while ensuring stable radiation characteristics, high antenna efficiency, and broadband impedance matching. Unlike existing beam-steering mechanisms in reflectarrays and antennas, Tables I and II, our proposed hemispherical transmitarray achieves wider steering angles through a curved aperture, multilayer electromagnetic interactions, and fine phase control using graphene sectors. Owing to the overlapping resonances of the FCC elements across adjacent rings and the stable electromagnetic response of the multilayer structure, our proposed transmitarray maintains broadband impedance matching, |S11 | < −10 dB with a relative bandwidth of 40% from 200 to 300 GHz, across all beam-steering directions. Compared with the available literature, Tables I and II, numerical results of the radiation patterns have verified that our proposed transmitarray achieves wide-angle 3D beam steering through accurate voltage-controlled transmission-phase tuning of 121 independently programmable graphene sectors. Our proposed hemispherical transmitarray antenna maintains high efficiency over a wide beam-scanning range. Even at the maximum steering angles, the antenna achieves efficiencies of 71% and 80%, representing less than an 8% drop from the broadside performance. This high performance stems from the low-loss multilayer structure, tunable graphene unit cells, and the intrinsic phase compensation of the hemispherical aperture. These features enable wide-angle beam steering with high aperture efficiency for THz beam- and target-tracking applications. R EFERENCES [1] O. Balci, E. O. Polat, N. Kakenov, and C. Kocabas, “Grapheneenabled electrically controlled terahertz plasmonic switches,” Nature Communications, vol. 6, p. 6628, 2015.

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