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Graphene-based Hemispherical Transmitarray Antenna for Wide-Angle Beam Steering and Ultrafast Moving Target Tracking
arXiv:2607.26437v1 [cs.NI] 29 Jul 2026
Somayeh Komeylian, Member, IEEE and Christopher Paolini, Member, IEEE
Abstract—This work expands the application of dynamically tunable graphene to implement a hemispherical transmitarray antenna tailored for wide-angle electronic beam steering and ultrafast moving-target tracking at 250 GHz. The proposed hemispherical transmitarray antenna features a dynamically reconfigurable graphene-based multilayer configuration consisting of gold radiating patches, biased graphene sectors, hBN dielectric layers, and a centrally positioned hornfeed. The analytical framework of graphene-based reconfigurable metasurfaces including graphene surface-conductivity modeling, voltage-controlled surface impedance, transmission-coefficient, and conformal array-factor analysis has been established for hemispherical transmitarray antenna for the first time. Moreover, the analytical and practical mapping between desired beam directions and the corresponding graphene bias voltages have been developed for the transmitarray antenna in the first time. Our transmitarray antenna possesses exceptional 3D beam steering, providing wide-angle elevation scanning from −87◦ to 87◦ and full 360◦ azimuthal coverage. It also exhibits an antenna efficiency ranging from 68% to 80% with a 10% degradation. Index Terms—Metasurface transmitarray antenna, reconfigurable metasurfaces, fractal concentric circular elements, voltagecontrolled graphene, anisotropic surface impedance, ABCD matrix, generalized sheet transition conditions, Huygens metasurfaces, adaptive beamforming, 6G wireless communications.
I. I NTRODUCTION
B
EAM steering is a critical feature of next-generation antenna systems for THz frequency regime, providing adaptive directional radiation for high-capacity wireless communications, radar, and remote sensing applications, table 1. Various advanced beam-steering antenna technologies, including phased-array antennas (PAAs), reflectarrays, near-field array antennas, and transmitarray/metalens antennas, have been extensively developed and investigated by numerous research groups, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Despite their excellent beam-steering performance, phased array antennas (PAAs) face several fundamental limitations associated with their conventional beamforming architectures, including high hardware complexity, substantial power consumption, and limited scalability for ultra-fast wide-angle scanning applications. Each antenna element typically requires an individual active phase 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: [email protected]). C. Paolini is with the Department of Electrical and Computer Engineering, San Diego State University, San Diego, CA 92182 USA (e-mail: [email protected]).
shifter, and in many implementations, additional amplifiers or attenuators, leading to a significant increase in their system complexity, cost, and power consumption. Furthermore, the obtainable scanning speed is constrained by the switching response of phase shifters and the processing latency of beamforming control electronics. Real-time beam steering in PAAs requires continuous calculation, synchronization, and adjustment of the phase and amplitude excitation coefficients for all array elements, further increasing computational and hardware demands, [12]. Conventional reflectarray antennas typically achieve beam steering through mechanical displacement of their feeds or multiple feed switching to modify the incident phase distribution, rather than through electronically reconfiguration of the phase response of individual unit cells, [17], [1]. Hence, mechanical feed displacements limit their scanning speeds to the millisecond-to-second range, [1], [2]. Even with multiple feedings, switching is generally restricted to discrete beam directions rather than continuous scanning. Furthermore, placing the feed antenna in front of the reflecting aperture causes aperture blockage, which drastically degrades efficiency, gain, and illumination uniformity, [17], [4]. This limitation becomes more pronounced at wide scanning angles, where severe phase errors, feed aberrations, and illumination nonuniformity exacerbate scanning loss and degrade beam quality, [12], [6]. As a result, the resonant nature of conventional unit cells strictly limits the operating bandwidth of many reflectarrays. This restriction limits the frequency range over which reflectarrays can maintain a stable phase response and high radiation efficiency, [18], [19]. Near-field metasurface beam-steering antennas achieve twodimensional beam reconfigurations by incorporating engineered metasurface layers above the radiating aperture, enabling spatial control of the radiated electromagnetic wavefront. However, the addition of multiple metasurface layers to enhance phase manipulation capability increases the antenna profile and fabrication complexity, thereby limiting low-profile integration, [6], [12]. Furthermore, electronically reconfigurable implementations often require complex biasing networks, while their obtainable scanning speed is limited by their response time of the tuning elements and associated control circuitry, [2], [3]. Moreover, the dielectric and conductor losses introduced by multilayer architectures can degrade radiation efficiency, particularly at millimeter-wave and terahertz frequencies, where material losses become increasingly significant, [4], [12]. Metalens antennas provide low-profile, high-integration
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solutions for beam focusing and steering by engineering the TABLE II: Performance comparison and categorization of statetransmission phase profile of metasurface elements, [3], [4]. of-the-art lens and metasurface antenna beam-reconfiguration However, conventional metalens antennas typically rely on fixed methodologies. phase profiles or mechanical feed displacement, which severely Beam Recon- Refs. Frequency Beam Scanning Scanning Main Range Steering Range Speed Limitations restricts real-time reconfigurability and limits scanning speeds, figuration Mechanism Method [1], [2]. Furthermore, single-focus metalenses compensate [3,5,8,10] 5.75–40 PIN diodes, ±30◦ – ns–µs High hardware only for a first-order approximation of the required wavefront Electronic Switching GHz varactors, RF ±60◦ complexity, switches, insertion loss, phase. This introduces substantial nonlinear phase errors at multi-feed and power large steering angles, causing severe scan loss, a reduction selection, consumption optimized in the realized gain, and a degradation in the beam quality phase coding during wide-angle operations, [12], [6], [8]. To overcome the ◦ TABLE I: Comparison of beam-steering mechanisms and scanning speeds. Rank Technology 1 2 3 4 5 6
Mechanical Beam Steering
[13]
450–650 GHz
Piezoelectric ±25 actuators translating a silicon lens or sparse lens array
Slow
Moving mechanical parts and limited long-term reliability
Passive FrequencyDependent
[11]
610–685 GHz
16◦ – Frequency32◦ dependent GRIN/Fresnel- FoV lens phaseresponse interposer
Fast
Beam direction strictly coupled to operating frequency
Passive
[6,9]
2.1–75
1-bit binary
Speed
Graphene voltage tuning (our work) ns–µs Frequency-dependent passive beamforming [7,11] ns–µs RF switch beam selection (LAS, mALL) [2,5,8,9] ns–µs Digital beamforming with phase shifters [2,10] ns–µs Mechanical feed displacement [4,13] ms–s Reconfigurable metasurface lens replacement [1,6] Offline (min)
±30◦ – ◦
Static/FastNo continuous
GHz metasurfaces, ±60 real-time beam aforementioned constraints, this work introduces a graphene- Fixed/Discrete fixed phase tracking based hemispherical transmitarray antenna in Fig. 1. The main distributions, DFrFT motivations and core contributions of this research are outlined networks in the following. Feed-Position [12] 24.25– Multi-focus ±48◦ ms–s Complex feed Ultrafast Graphene-based Beam Steering and Real- Multi-Focus 25.8 GHz phase configuration compensation and limited Time Target Tracking: Our proposed graphene-based transusing flexibility mitarray antenna is specifically tailored for millimeter-wave feed-position switching (mmW) and the THz regime, overcoming the high propagation Ultrafast: Not reported This 200–300 Mapping ±87◦ loss, narrow tuning bandwidths, and hardware complexity VoltageControlled work GHz desired beam elevans–µs of conventional beam-steering antennas, Table II. A major Graphene direction to tion and ◦ graphene bias 360 advantage of graphene relies in its electrically tunable surface Tuning voltages azimuth conductivity, enabling real-time modulation of the unit-cell surface impedance and phase response without incorporating conventional semiconductor active devices into the terahertz electromagnetic structure. By independently tuning the chemi- required 11 × 11 voltage-control matrix (or retrieves it from a cal potential of the isolated graphene sectors, our transmitarray precomputed lookup table) for a specified beam-steering angle antenna achieves programmable phase and amplitude control, and distributes the corresponding bias signals to multiple highwhich allows for rapid electronic beam reconfiguration without speed multichannel digital-to-analog converters (DACs) via mechanical scanning or feed displacement, Table I. This agile parallel serial peripheral interface (SPI) or low-voltage differbeam adaptation is highly advantageous for emerging 6G ential signaling (LVDS) interfaces. The proposed row-parallel communication systems, high-resolution radar, and remote control architecture enables full-aperture reconfiguration within sensing systems that require rapid responses to dynamic 10 − 20µs, corresponding to 50,000-100,000 beam updates propagation environments. Moreover, the fast reconfiguration per second. This scalable implementation avoids the excessive capability, potentially reaching the nanosecond-to-microsecond hardware overhead and complexity of deploying an individual regime and primarily determined by the graphene biasing DAC channel for every graphene element while maintaining network rather than mechanical limitations, enables rapid the high-speed reconfigurability required for adaptive beam adjustment of the aperture phase distribution and beam direction. steering and real-time target tracking, Table I. Consequently, the proposed antenna can maintain continuous High Gain and Broadband Hemispherical Transmitarray beam alignment and compensate for angular displacements of Aantenna: From a structural perspective, the hemispherical moving targets over a wide steering range, Figs. 9 and 10. curvature of our antenna establishes an inherent phase-matching In this work, the proposed transmitarray antenna will be interface with the spherical wavefront emitted by the central integrated with the AMD Xilinx Zynq UltraScale+ RFSoC feedhorn, thereby mitigating aperture phase errors and enhancZCU670 platform available in our laboratory to enable real- ing radiation efficiency, Table IV. Furthermore, scaling the time tracking of moving targets through dynamic beam dimensions of the fractal concentric circular (FCC) elements, steering. A practical and scalable bias-control implementation Fig. 2, from row to row excites multiple coupled resonant is achieved by organizing the 121 graphene elements into modes with closely spaced frequencies. This yields overlapping an 11 × 11 voltage-control matrix. The FPGA computes the resonances that significantly expand the operational bandwidth,
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Figs. 3 and 4. II. G EOMETRIC C ONFIGURATION AND D ESIGN Wide Spherical Coverage and Wide-Angle Beam Steering PARAMETERS OF THE METASURFACE TANSMITARRAY for 6G Wireless Communications: Spherical coverage defines A NTENNA the solid-angle range over which an antenna effectively radiates In this section, the design topology of the proposed metaor receives electromagnetic energy, [17]. As wireless networks surface transmitarray antenna is first presented. The stacked transition toward the THz spectrum for 6G communications, multilayer arrangement of our transmitarray antenna is inpropagation conditions become increasingly challenging due troduced, and the functional role of each constituent layer, to severe free-space path loss and atmospheric absorption. including the electrically tunable graphene layer, metallic FCC Although antenna arrays are traditionally deployed to mitigate patch layer, hBN dielectric isolation layers, gate dielectric these losses through high directional gain, such high-gain layer, and supporting substrate, is discussed in detail. In configurations inherently restrict the radiation beamwidth, the following, the key geometric design parameters of our and thereby angular coverage. Consequently, the achievable proposed transmitarray antenna are systematically investigated, spherical coverage is significantly influenced by the antenna as they critically determine the electromagnetic behavior of topology, beam-steering capability, and the interference char- individual unit cells by controlling their resonant characterisacteristics of the wireless communication channel, [7], [11], tics, impedance response, and wave transmission properties. [20], [4]. To address these limitations, our proposed three- Accordingly, this section investigates the geometric parameters dimensional hemispherical transmitarray antenna provides a of the gold FCC patches and graphene sectors, including vastly superior spherical coverage compared to conventional their spatial distribution, dimensional scaling, and ring-to-ring planar configurations, [7], [11], [20], [4]. Its curved architecture dimensional variation of the unit cells across the hemispherical enables wide-angle beam steering over an broad field of lens aperture. It is important to note that, although the graphene view while maintaining high directive gain, Figs. 9 and 10, sectors and metallic FCC patches are spatially aligned with thereby elevating the average signal-to-interference ratio (SIR). coincident centers, they remain electrically isolated through an Consequently, our antenna offers enhanced robustness against intermediate hBN dielectric layer. This insulating hBN layer rapid channel variations, positioning it as a compelling solution prevents direct electrical contact between the two conductive for future THz-band 6G wireless communication networks. materials, thereby avoiding unintended short circuits while An Electromagnetic Modeling Framework for a Voltage- preserving independent electromagnetic control of each unit Reconfigurable Graphene Hemispherical Transmitarray cell, as illustrated in Fig. 1. Antenna: To the best of the authors’ knowledge, this work is the first to establish a unified electromagnetic analytical framework for a multilayer graphene-based hemispherical transmitarray antenna. The framework analytically derives formulations for the voltage-controlled surface impedance, equivalent lumped RLC circuit model, transmission coefficient of S21 , antenna efficiency, quasi-active spherical array factor, and the mapping between desired beam directions and the corresponding graphene bias voltages, thereby enabling the systematic design and analysis of ultrafast beam-steering systems for real-time moving-target tracking. Consequentially, the remainder of this work is organized into the eight following sections: Section II describes the design parameters and geometric configuration of the proposed hemi- Fig. 1: The hemispherical transmitarray antenna comprises spherical graphene-based metasurface transmitarray antenna. 121 patch elements distributed over the hemispherical aperture, Section III develops equivalent impedance models, including including one apex element and five concentric rings containing voltage-controlled graphene conductivity, multilayer equivalent 8, 16, 24, 32, and 40 patches, respectively. A geometric scaling circuit, and anisotropic surface impedance formulation. Section factor of 0.8 is applied to the patch dimensions from one ring to IV is devoted to the analytical formulation for extracting the the next. Our stacked multilayer transmitarray antenna consists transmission coefficient of S21 of the multilayer transmitarray of SiO2 substrate with a thickness of 0.5 mm, followed by unit cell. Section V establishes the analytical and practical map- the first hBN dielectric layer with a thickness of 0.5 mm, ping between the desired beam direction and the corresponding the graphene sectors, the second hBN dielectric layer with a bias-voltage distribution applied to the graphene sectors. In thickness of 0.5 mm, and FCC patches made of gold (Au) on section VI, we have derived the conformal hemispherical array the top layer exposed to free space. factor and the beamforming formulation. The functionality and 1) Stacked Multilayer Configuration of the Transmitarray performance of our proposed metasurface transmitarray antenna have been verified and validated in section VII. Section VIII Antenna: The multilayer stack configuration of the our transevaluates the transmission efficiency and the overall antenna mitarray antenna, Fig. 1, consists of the following layers. efficiency of the proposed hemispherical transmitarray antenna. FCC gold pathes: The gold FCC patch frequency-selective Finally, Section IX concludes this work and summarizes the surface (FSS) acts as the transmitting resonant layer and principal contributions. controls the amplitude and phase response of each unit cell. This
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layer, including gold FCC patches, radiates into the surrounding free-space region (outside air). The Random Hill-Climbing (RHC) framework has been extensively employed to improve the operating bandwidth of antennas or reflectors by optimizing row-dependent resonant elements with different geometrical dimensions,[19], [21], [22]. RHC model is a local optimization algorithm that iteratively maximizes an objective function. Starting from an initial random or predefined solution, it generates neighboring solutions by modifying the binary representation of the current solution and retains any candidate that improves the objective function. This work adopts the objective function proposed in [18] to generate optimized FCC patches Fig. 2. The resulting FCC patches are subsequently mapped onto the hemispherical transmitarray lens, thereby significantly enhancing the antenna bandwidth. In this work, the objective is to determine the
Fig. 2: The view of a fractal concentric circular patch of our proposed metasurface antenna in Fig 1. The FCC patch element diameter is reduced by a geometric scaling factor of 0.8 from each inner ring to the adjacent outer ring, [22]. values of α that maximize the useful S21 transmission-phase tuning range over the allowable chemical-potential interval µc = 0.1–1 eV while maintaining sufficiently high transmission magnitude (i.e., low insertion loss). Hence, by applying the aforementioned condition, the gold patch scaling factor per ring is determined to be α = 0.8 reasonable with a choice of a = 8. Indeed, in our hemispherical transmitarray antenna, the dimensions of the FCC elements scale by a constant factor of 0.8 from one ring to the next, as detailed in Table I of [16]. It is evident in Fig. 3 that a geometric scaling factor of 0.8 was determined through parametric optimization to maximize the transmission coefficient of S21 . The unit-cell dimensions are gradually varied from one ring to the next along the radial direction of the hemispherical aperture with the scaling factor of α. This causes each concentric ring to resonate at a distinct frequency that is closely spaced to those of its neighboring rings, resulting in overlapping resonances and a significant enhancement of the overall operating bandwidth. The superposition of these adjacent resonances broadens the overall transmission and impedance bandwidth. This is noticeable in Fig. 4 that our hemispherical transmit-array antenna achieves an exceptionally wide operating bandwidth while fully preserving its beamforming capability. hBN1 : The first hBN layer serves as ultra-thin dielectric isolation layers and gate dielectrics, preventing direct electrical contact between graphene and metallic elements while enabling
Fig. 3: A demonstration of the geometric scaling factor of the FCC patches was optimized, and a value of 0.8 was selected to maximize the transmission coefficient (S21 ) of the unit cell.
Fig. 4: S11 variations in terms of frequency in GHz for our metasurface transmitarray antenna consistent with Fig. 1, reflecting an exceptionally wide bandwidth for an operating frequency of 250 GHz and at θ = 0◦ and ϕ = 0◦ .
efficient electrostatic tuning of graphene conductivity. The hBN1 layer electrically isolates the graphene layer from the metallic patches and acts as a dielectric spacer, while the graphene layer provides voltage-controlled tunability through variation of its surface conductivity. hBN1 electrically isolates the graphene layer from the metallic FCC patches, preventing unintended short circuits while providing efficient capacitive coupling for electrostatic tuning. Graphene sectors: First, the continuous graphene layer is discretized into electrically isolated trapezoidal sectors, each corresponding to a single FCC element, thereby enabling independent voltage biasing and programmable electromagnetic control. The FCC gold patches are centrally aligned with their corresponding graphene unit cells to maximize the electromagnetic interaction between the resonant metallic structure and the tunable graphene layer. This arrangement enhances the achievable transmission-phase control while preserving independent electrical tuning through the intermediate hBN dielectric layer. The graphene sectors introduce a dynamically reconfigurable surface impedance by varying the graphene chemical potential through applied bias voltage, thereby allowing independent tuning of the transmission phase and amplitude responses of the metasurface patches. The applied voltage bias modifies the graphene carrier concentration, which in turn dynamically
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adjusts its surface conductivity and provides tunable control of the electromagnetic response and transmission phase of each unit cell. hBN2 : An alternative fabrication strategy is proposed to preserve the integrity of the SiO2 substrate by avoiding direct surface etching. In this approach, the second uniform hBN layer of hBN2 is first deposited on the SiO2 substrate, and the required microchannels are subsequently patterned and formed exclusively within this hBN layer. During fabrication, microchannels are designed to enable independent voltage biasing and dynamic control of each graphene sector while maintaining the mechanical integrity and low-loss electromagnetic properties of the supporting substrate. Therefore, the microchannel fabrication process is confined to the hBN layer rather than being performed directly on the SiO2 substrate, thereby preserving the inherent low-loss characteristics and high-quality electromagnetic performance of the proposed metasurface structure. SiO2 lens: The SiO2 layer provides the electromagnetic interface between the horn excitation and the reconfigurable metasurface layers. It mechanically supports our conformal metasurface antenna, introduces controlled dielectric loading, and preserves the hemispherical geometry required for the radial transformation of the transmitted electromagnetic wavefront. Circular feedhorn: The excitation horn is positioned at, or very near, the geometric center of the hemisphere, ensuring that the incident electromagnetic field illuminates the hemispherical aperture predominantly along the local radial direction. This configuration minimizes phase variation across the aperture and enables efficient coupling between the feed source and the conformal metasurface elements. The horn antenna serves as the high-frequency RF excitation source, while the graphene bias network provides low-frequency electrostatic tuning. The biasing circuitry is therefore considered a control mechanism rather than an RF feeding structure and should not be included in the electromagnetic excitation path. 2) Gold Patch Dimensions and Geometric Scaling Analysis: The electromagnetic characteristics of our transmitarray antenna are substantially affected by the geometric distribution of gold patches across the hemispherical aperture. Our proposed active metasurface is conformal to a hemispherical lens with the radius of R. Specifically, a, n, α mainly govern the spatial phase sampling of the unit cells over the hemispherical aperture. They control the number of unit cells per ring, the local azimuthal period of D(n,k),ϕ , the effective area of each unit cell, ∆Sn , and the azimuthal coordinate of each unit cell of ϕ(n,k) . Subsequently, these spatial dimensions determine the local graphene surface impedance of Zg,(n,k),ϕ , and the equivalent admittance of the gold-patch FSS of YAu,(n,k),ϕ . On the other hand, the scaling parameter of α regulates the dimensions of the resonant metallic structures. Specifically, the goldpatch length, LAu,(n,k),ξ , and the corresponding inter-patch gap width, dAu,(n,k),ξ , specifically, the gold-patch length of LAu,(n,k),ξ , and their corresponding inter-patch gap width of dAu,(n,k),ξ , is explicitly determined by α. Hence, these geometrical parameters control the equivalent gold-patch FSS admittance of YAu,(n,k),ξ , which is accompanied by regulating the magnitude and phase of the local transmission coefficient of
S21,(n,k) . Consequently, whereas a and the distances between unit cells define the spatial sampling grid and unit cell distribution across the hemispherical aperture, α controls the local electromagnetic response by tuning the characteristics of the gold-patch FSS. The functional dependencies between these design parameters and the resulting electromagnetic quantities are summarized as follows, n (a, ρ) −→ Nn , Dϕ,(n,k) , ∆S(n,k) , ϕ(n,k) , o (1) Zg,(n,k),ξ , YAu,(n,k),ξ n o α −→ LAu,(n,k),ξ , dAu,(n,k),ξ , YAu,(n,k),ξ , S21,(n,k)
(2)
ξ ∈ {θ, ϕ} where ξ ∈ {θ, ϕ} represents the polarization index, corresponding to the θ- or ϕ-polarized electromagnetic field component. Consequently, the local surface impedance, the local transmission coefficient, and the overall spherical array factor are formulated explicitly as functions of the design parameter vector, p = {a, α, Nr , R, θmax , Vn,k , f, . . .}
(3)
where R, V(n,k) , and f denote the hemisphere radius, a bias voltage applied to the (n, k)-th graphene sector, and the operating frequency, respectively. θmax represents the maximum polar angle covered by the hemispherical aperture. The analytical and algorithmic optimization framework is governed by these geometrical parameters, enabling the hemispherical transmitarray antenna to realize adaptive beam steering, and thereby improving transmission efficiency. The maximum element size is selected as Dn,max = 0.5λn,max , while the spacing between adjacent elements within each row is fixed at 0.15λn,max , where λn,max denotes the wavelength corresponding to the resonant frequency of the n-th ring. 3) Angular Discretization and Geometric Arrangement of Graphene Sectors: Assume that a uniform graphene layer lies on a hemisphere with radius R. The position vector of the center of the (n, k)-th graphene sector, r(n,k) = R sin θn cos ϕ(n,k) x̂ + sin θn sin ϕ(n,k) ŷ + cos θn ẑ (4) To enable independent electromagnetic control of each FCC patch element, an individual bias voltage is assigned to each graphene sector. Hence, the continuous graphene layer is therefore discretized into electrically isolated trapezoidal sectors. Indeed, the continuous graphene layer deposited on the hemispherical surface is discretized into electrically isolated trapezoidal sectors using the angular intervals of ∆θ and ∆ϕ. The dimensions of these sectors along the two orthogonal tangent directions are derived directly from the spherical coordinate system. From the mathematical derivation, the elevation position of (n, k)-th graphene sectors are given by, 1 θn = n − ∆θ, n≥1 (5) 2 ∆θ =
θmax Nr
(6)
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the spherical curvature. Near the apex of the hemisphere (θn → 0), the circumference of each ring approaches zero, resulting in progressively smaller azimuthal cell periods and a correspondingly denser spacing of the graphene sectors. Conversely, toward the equatorial region (θn → π/2), the available circumference increases, requiring a larger number of sectors to maintain approximately uniform spatial sampling and conformal coverage over the hemispherical aperture. Dθ,(n,k) = R∆θ
Fig. 5: The hemispherical transmitarray consists of 121 graphene sectors distributed over the hemispherical surface, where each locally planar cell is approximated by a trapezoidal geometry. Each graphene sector is independently controlled by a distinct gate voltage corresponding to a graphene chemical potential ranging from 0.1 to 1.0 eV, while all cells share a common carrier relaxation time τ = 0.1ps and an operating temperature of T = 300K. All the graphene sectors are physically and electrically isolated from adjacent sectors to ensure independent electrostatic biasing and stable reconfigurable operation. where θmax denotes the maximum polar angle of the active hemispherical aperture measured from the positive z-axis, which is equal to π/2 for a complete hemispherical aperture. The azimuthal position of the (n, k)-th unit cell is given by,
(11)
where ∆θ represents the angular spacing between consecutive rings. For a uniformly discretized hemispherical aperture, this meridional period remains constant for all rings because it is independent of the polar angle θ(n,k) . Hence, the approximate area of each sector on the hemispherical curvature is given by, 2π = R2 sin θn ∆θ ∆ϕn (12) Nn 2π ∆ϕn = (13) Nn The graphene sectors are designed to be electrically isolated from one another, thereby enabling independent tuning of the surface impedance of each unit cell through separate gatevoltage biasing. In this sense, adjacent graphene cells should be separated by finite and physical gaps in the following expressions, ∆S(n,k) ≈ R2 sin θn ∆θ
Gθ,(n,k) = Dθ,(n,k) − gθ,(n,k) Gϕ,(n,k) = Dϕ,(n,k) − gϕ,(n,k)
(14)
Unlike a planar periodic array, the period in the ϕ-direction 2π(k − 1) , k = 1, 2, . . . , Nn (7) depends on θ. Near the apex of the hemisphere, sin θ is small, Nn and therefore the sectors become more compressed along the where ϕ0,n represents the initial azimuthal offset of the (n, k)- azimuthal direction. th ring that can be introduced to stagger adjacent rings and improve the spatial distribution of the graphene sectors. Nr III. VOLTAGE -C ONTROLLED S URFACE I MPEDANCE AND and Nn represent the number of rings, excluding the apex T RANSMISSION M ODELING OF THE G RAPHENE - BASED unit cell, and the number of unit cells distributed on each ring, T RANSMITARRAY A NTENNA respectively. The local cell period along the azimuthal direction The electromagnetic properties of the graphene layers are is determined by the circumferential arc length of each ring characterized by using the Kubo formulation. The graphene and varies with the polar position on the hemisphere by the sheet is modeled as an infinitesimally thin conductive surface following form, with a surface impedance defined as, [23], 2πR sin θ(n,k) 1 Dϕ,(n,k) = R sin θ(n,k) ∆ϕ = (8) Zg (ω, µc ) = (15) Nr σg (ω, µc ) For a linear ring-population model, the number of graphene where σg represents the complex surface conductivity of the sectors distributed along the n-th ring is expressed as, graphene. For the low-terahertz frequencies considered in this Nr = a [1 + β(n − 1)] (9) work, the graphene conductivity is dominated by the intraband where a represents the initial number of unit cells in the contribution of the Kubo formulation and can therefore be first ring and β denotes the ring-to-ring growth coefficient. approximated by the Drude model, ϕ(n,k) = ϕ0,n +
The element distribution for the first five rings, with Nn = [8, 16, 24, 32, 40, 48, 64, 80], is exactly described by Eq.9 using a = 8 and β = 1, resulting in the linear relation Nr = 8n. The local azimuthal period in n-th ring becomes, 2πR sin θ(n,k) Dϕ,(n,k) (a, β) = (10) a [1 + β(n − 1)]
The corresponding DC conductivity, σ0 , is given by, [23], µc e2 kB T τ − kµcT B σ0 = + 2 ln 1 + e (17) πℏ2 kB T
in contrast to the constant meridional period, the azimuthal period is substantially dependent on the polar angle due to
where e, kB , T , ℏ, τ and µc represent the elementary charge, the Boltzmann constant, the absolute temperature, the reduced
σg (ω, µc ) =
σ0 (µc ) 1 + jωτ
(16)
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Planck constant, the carrier relaxation time, and the graphene chemical potential respectively. For the biasing conditions considered in this work, where the graphene chemical potential satisfies µc ≫ kB T , the Kubo formulation simplifies to the following Drude expression, σg (ω, µc ) ≈
e2 µc 2 πℏ (τ −1 + jω)
(18)
where 1/τ denotes the carrier scattering rate. Considering the periodic distribution of graphene sectors over the hemispherical transmitarray aperture, the equivalent surface impedance is formulated by incorporating the intrinsic graphene conductivity as well as the reactive loading arising from the unit-cell geometry and inter-element coupling as follows, j D − Zg = (D − d)σg ωCeff
(19)
surface concept, [25], [26], to determine the local voltagecontrolled anisotropic impedance of the graphene sectors and gold patches. This modeling approach decomposes the overall electromagnetic response of the transmitarray antenna into two complementary parts: (1) the local unit-cell response, in which the graphene sectors exhibit tunable RLC characteristics similar to planar metasurface elements, and (2) the global hemispherical electromagnetic response is incorporated through the spherical arrangement of the unit cells, the ABCD-matrix formulation of the multilayer transmission system, and the corresponding S21 transmission characteristics across the curved aperture. A. Voltage-Controlled Surface impedance of Graphene Sectors The graphene layer, consistent with Fig. 5, is modeled through the Kubo conductivity, σ(µc ), which captures the dependence of the sheet conductivity on the chemical potential. Indeed, the tunability of graphene originates from the variation of its complex surface conductivity with the chemical potential, µc . To incorporate this effect into the electromagnetic model, the graphene layer is represented as an equivalent conductive sheet characterized by the surface admittance Ys . The relationship between the transmission coefficient and the sheet admittance of a zero-thickness conductive sheet is given by,
where D and d refer to the unit-cell period and the graphenesector dimension, respectively. The first term of Eq.19 represents the scaled material impedance of the graphene patches, while the second term accounts for the reactive contribution of the fringing electric fields across adjacent sectors. Hence, for the periodic arrangement of graphene sectors over the hemispherical transmitarray aperture, the equivalent surface impedance described in Eq.19 incorporates both the intrinsic graphene conductivity and the geometry-induced reactive 2 T = (23) effects of the periodic unit-cell structure. The effective gap 2 + Y s Z0 capacitance in Eq. 20 is approximated using the classical closedwhere Z0 is the free-space impedance. Accordingly, the form FSS expression, [24], equivalent sheet admittance, which fully captures the bias πd ε0 (εr + 1)D ln csc (20) dependent electromagnetic response of graphene, is obtained Ceff = π 2D as, 2(1 − T ) where ϵ0 and ϵr denote the vacuum permittivity and the relative (24) Ys = Z0 T permittivity of the underlying substrate, respectively. The surface conductivity is dynamically adjusted via electrostatic and the corresponding sheet impedance is given by, biasing. Therefore, for the (n, k)-th unit cell, the electrically 1 Zs = (25) induced chemical potential µc,(n,k) is determined by the applied Ys gate voltage V(n,k) according to, To achieve low-reflection transmission with full phase control, s the unit cell is modeled using the generalized sheet transiπCg V(n,k) − VD µc,(n,k) = ℏvF (21) tion conditions (GSTCs) or the equivalent Huygens’ surface e representation, [25], [26]. The GSTC model represents the where vF , Cg , and VD represent the Fermi velocity, the gate finite-thickness multilayer unit cell by an equivalent zerocapacitance per unit area, and the Dirac voltage, respectively. thickness Huygens surface that preserves the same transmission This local voltage modulation establishes a spatially tuning and reflection characteristics. Unlike a purely electric sheet conductivity profile, approximation, the Huygens model accounts for both electric σg,(n,k) = σg ω, µc,(n,k) (22) and magnetic surface responses generated by the combined interaction of the graphene sectors, gold FCC patches, hBN Consequently, adjusting the localized gate voltage directly dielectric layers, and the supporting silicon substrate. For a controls the graphene sheet conductivity, thereby modulating transmissive metasurface supporting both electric and magnetic the equivalent surface impedance of each individual transmi- surface responses, the corresponding boundary conditions are tarray unit cell. This voltage-controlled impedance provides expressed as, independent, reconfigurable tuning of the transmitted wave’s n̂ × (H2 − H1 ) = Yse Eav (26) amplitude and phase, enabling dynamic beam steering and adaptive wavefront manipulation in our proposed hemispherical (E2 − E1 ) × n̂ = Zsm Hav (27) graphene-based transmitarray antenna. Accordingly, this section develops the equivalent surface impedance model for the When only the graphene sheet is considered, the metasurface multilayer transmitarray antenna. We use the generalized sheet response is represented solely by an electric surface admittance. transition condition (GSTC) formulation and the Huygens Since an infinitesimally thin graphene layer does not exhibit
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an intrinsic magnetic surface response, the equivalent magnetic Hence, the corresponding graphene surface impedance comsheet impedance is assumed to be zero, i.e., Zsm = 0. ponents along the meridional and azimuthal directions are Accordingly, the GSTC formulation reduces to the electric sheet expressed as, boundary condition, where Yse represents the equivalent electric D(n,k),θ j − (34) Zg,(n,k),θ = sheet admittance of the graphene layer, consistent with graphene G(n,k),θ σg,(n,k) ωCg,(n,k),θ sectors in Fig. 5. It is worth mentioning that the magnetic response does not originate from the graphene layer alone; The equivalent gap capacitances along the two orthogonal rather, it emerges from a combination of the electromagnetic tangential directions are approximated using the closed-form interactions associated with the stacked multilayer unit cell, FSS formulation as, [27], [23], including the graphene sectors, metallic FCC patches, dielectric ε0 εeff D(n,k),θ πg(n,k),θ Cg,(n,k),θ = ln csc (35) layers, and supporting substrate. Therefore, the equivalent π 2D(n,k),θ electric surface admittance and magnetic surface impedance should be represented as second-order tensors to account for Consequently, the explicit expressions demonstrate that the the anisotropic response of the stacked multilayer elements in local graphene surface impedance is jointly determined by the the local spherical coordinate system. Accordingly, the electric voltage-controlled graphene conductivity, the anisotropic unitcell geometry, and the direction-dependent gap capacitances. surface admittance tensor is expressed as, Y 0 Yse = θθ (28) B. Gold-Patch Local Sheet Admittance 0 Yϕϕ The gold layer is modeled as a ring-dependent FSS, whose while the equivalent magnetic surface impedance tensor is equivalent sheet admittance captures the electromagnetic regiven by, sponse of the metallic pattern for each concentric ring. When m the unit-cell dimensions are electrically small relative to the Zθθ 0 Zsm = (29) operating wavelength, the electromagnetic response is predomim 0 Zϕϕ nantly capacitive due to the strong fringing electric fields across For reflectionless transmission with a desired (or required) the inter-patch gaps. Under this first-order approximation, the phase response of ψ, the transmission coefficient is defined as, sheet admittance along the two orthogonal tangential directions is expressed as, T = ejψ (30) YAu,(n,k),θ = GAu,(n,k),θ + jωCAu,(n,k),θ (36) Hence, the corresponding equivalent electric and magnetic YAu,(n,k),ϕ = GAu,(n,k),ϕ + jωCAu,(n,k),ϕ (37) surface parameters for a reflectionless Huygens transmissive sheet with a desired phase response ψ can be derived as, Here, GAu represents the conductive loss of the metallic FSS, while the direction-dependent capacitances describe the 2j ψ Yse = − (31) anisotropic electromagnetic coupling between adjacent gold tan Z0 2 FCC patches over the hemispherical transmitarray surface. For the periodic gold-patch array, Fig. 1, the equivalent gap ψ Zsm = −2jZ0 tan (32) capacitances along the two orthogonal tangential directions 2 are approximated using the well-established closed-form FSS capacitance formulation reported in [27] and [23], These last equations indicate that a purely electric sheet ε0 εeff,Au D(n,k),θ πdAu,(n,k),θ cannot generally provide complete transmission-phase coverage CAu,(n,k),θ = ln csc while preserving high transmission efficiency. In contrast, a π 2D(n,k),θ (38) combination of graphene, hBN, gold patch, and dielectric lens can behave as an effective Huygens-like transmitarray unit cell, A commonly used first-order approximation for the effective enabling efficient and independent control of the transmitted permittivity is given by, wave amplitude and phase. ϵupper + ϵlower ϵeff ≈ (39) However, for the multilayer transmitarray unit cell, the 2 ABCD-matrix formulation provides a more rigorous repre- Accordingly, when the graphene sheet is located at the interface sentation by explicitly modeling the cascaded electromagnetic between an hBN layer and a silicon substrate, the effective interactions of the graphene sectors, hBN dielectric layers, permittivity can be approximated as, gold FCC patches, and silicon substrate, thereby accurately ϵhBN + ϵSi predicting the overall transmission and reflection characteristics. ϵeff ≈ (40) 2 Because the unit-cell periods along the θ- and ϕ-directions differ, the local tangent-plane graphene surface impedance Conversely, when the graphene sheet is sandwiched between is inherently anisotropic. Accordingly, the graphene surface two hBN dielectric layers, the effective permittivity reduces to, ϵef f ≈ ϵhBN . However, when the FCC patches are not impedance tensor is expressed as, electrically small or when the operating frequency approaches (n,k) (n,k) Z g,(n,k) = Zg,θ θ̂ θ̂ + Zg,ϕ ϕ̂ϕ̂ (33) the intrinsic resonance of the FSS element, the capacitive
9
approximation alone is no longer sufficient. In this regime, the finite current path on the metallic patches introduces an inductive response, and the interaction between the inductive and capacitive effects produces the resonant behavior of the unit cell. Consequently, the equivalent sheet admittance is more accurately represented by an RLC circuit model, YAu,(n,k),ξ = GAu,(n,k),ξ + jωCAu,(n,k),ξ +
1 jωLAu,(n,k),ξ (41)
The capacitance term of the surface impedance in Eq.43 is decomposed into two components of a constant geometric capacitance, determined by the physical dimensions of gaps between adjacent graphene sectors on the curved surface, and a variable quantum capacitance, which varies with the applied bias voltages. Geometric capacitance: the fringing-field capacitance between adjacent patches on the curved surface of our hemispherical antenna is expressed by the following equation, Dθ Dϕ Gξ
In this model, GAu accounts for the finite conductivity of the gold patches, CAu represents the fringing-field capacitance between adjacent patches, and LAu models the inductive current flow along the finite metallic FCC elements. This RLC model accurately captures the resonant electromagnetic behavior of the gold FSS, thereby providing a more reliable description of the transmission amplitude and phase over a wide operating bandwidth.
Quantum capacitance: The quantum capacitance per unit area of the graphene sector is expressed as follows,
C. Equivalent Anisotropic Surface Impedance of the Multilayer Unit Cell
The absolute quantum capacitance is obtained by multiplying the per-unit-area capacitance by effective areas of graphene sectors,
Consider the complete transmitarray unit cell consisting of a graphene sector integrated with its corresponding multilayer stack, including the SiO2 , hBN dielectric layers and gold FCC patch. The complete unit cell is assumed to be as an equivalent electric sheet embedded in a symmetric medium. In this case, the impedance tensor of each multilayer stack unit cell is formulated in the spherical coordinate system and is characterized by its dependence on the applied graphene bias voltage in the following equation, Zs,(n,k),θ (Vg ) 0 Zs,(n,k) (Vg ) = (42) 0 Zs,(n,k),ϕ (Vg ) in which each polarization component is derived by the exact Drude–RLC equivalent circuit in Eq.43, Zs,(n,k),ξ (Vg ) = R(n,k),ξ (Vg ) + jωL(n,k),ξ (Vg ) 1 + jωCeff,(n,k),ξ (Vg )
(43)
The resistance term representing the spatial dispersion component of the surface impedance in Eq. 43 is given by, Dξ ′ 1 R(n,k),ξ = Re (44) Gξ σs,(n,k)
Cgeo,(n,k),ξ = ε0 εsub
Cq,(n,k) =
(48)
2e2 µc,(n,k) πℏ2 vf2
(49)
A = Dθ Dϕ
(50)
yields, Cq,abs,(n,k) =
2e2 Dθ Dϕ µc,(n,k) πℏ2 vf2
(51)
Consequently, because the quantum capacitance acts in series with the geometric capacitance, the total effective gap capacitance is given by, 1 Ceff,(n,k),ξ (Vg )
=
1 Cgeo,(n,k),ξ
+
1 Cq,abs,(n,k)
(52)
which can be simplified as, Ceff,(n,k),ξ (Vg ) =
Cgeo,(n,k),ξ Cq,abs,(n,k) Cgeo,(n,k),ξ + Cq,abs,(n,k)
(53)
Finally, the final lumped surface impedance as a function of Vg is obtained by substituting the voltage-dependent resistance, inductance, and effective capacitance into the equivalent RLC circuit and yields the complete voltage-controlled surface impedance,
or equivalently, Dξ ′ 1 + ω 2 τ 2 R(n,k),ξ (Vg ) = Gξ σ0,(n,k) (Vg )
(45)
Zs,(n,k),ξ (Vg ) =
where
e2 µc,(n,k) (Vg ) τ (46) πℏ2 Eq. 46 refers to the DC Drude conductivity. The corresponding inductance, L, is defined as, σ0,(n,k) (Vg ) =
L(n,k),ξ (Vg ) = τ R(n,k),ξ (Vg )
Substituting the explicit expression for the effective capacitance in Eq. 54,
(47)
where ξ ′ denotes the orthogonal periodicity (e.g., for Rθ , Dξ′ = Dϕ ).
Dξ′ τ Dξ ′ 1 + ω 2 τ 2 + jω Gξ σ0,(n,k) Gξ σ0,(n,k) j − ωCeff,(n,k),ξ (Vg ) (54)
Ceff,(n,k),ξ (Vg ) =
1 Cgeo,(n,k),ξ
+
πℏ2 vf2
!−1
2e2 Dθ Dϕ µc,(n,k) (55)
10
Substituting these constituent terms into the equivalent circuit model yields the lumped surface impedance explicitly as a function of the gate voltage, Dξ ′ 1 + ω 2 τ 2 τ Dξ ′ + jω − Zs,(n,k),ξ (Vg ) = Gξ σ0,(n,k) Gξ σ0,(n,k) ! πℏ2 vf2 j 1 + 2 ω Cgeo,(n,k),ξ 2e Dθ Dϕ µc,(n,k) (56)
dielectric capacitive impedances, the total unit-cell impedance becomes, Zcell (Vg ) = Rg (Vg ) + jωLg (Vg ) 1 1 1 + + + jωChBN2 jωCSiO2 jωCg
(60)
where Lg (Vg ), which is identical to L(n,k),ξ (Vg ) in Eq.57, denotes the inductance associated with the graphene sheet impedance. Accordingly, the impedance contributions of the SiO2 substrate and upper hBN dielectric layer are given by, η0 then the complete voltage-controlled surface impedance be√ tan k0 εr,SiO2 tSiO2 ZSiO2 = j √ (61) comes, εr,SiO2 Dξ′ 1 + ω 2 τ 2 and τ Dξ ′ η0 √ +jω Zs,(n,k),ξ (Vg ) = ZhBN2 = j √ tan k0 εr,hBN thBN2 (62) Gξ σ0,(n,k) Gξ σ0,(n,k) ε r,hBN | | {z } {z } Rξ (Vg )
Lξ (Vg )
1
j − ω
Cgeo,(n,k),ξ |
+
πℏ2 vf2
!
2e2 Dθ Dϕ µc,(n,k) {z } 1 Ceff,ξ (Vg ) (57)
or, equivalently, Zs,(n,k),ξ (Vg ) = R(n,k),ξ (Vg ) + jωL(n,k),ξ (Vg ) j − ωCeff,(n,k),ξ (Vg )
(58)
This formulation rigorously accounts for graphene geometry, substrate permittivity, carrier scattering time, kinetic inductance, and voltage-dependent quantum capacitance. Consequently, the surface impedance of each graphene sector is directly modulated by the applied gate voltage of Vg via its influence on the graphene chemical potential. This dependency establishes the physical framework for a tunable electromagnetic response, enabling dynamic phase control and adaptive beam steering in our hemispherical transmitarray antenna.
where η0 and k0 refer to the free-space impedance and the freespace wavenumber, respectively. t(SiO2 ) and t(hBN2 ) denote the thicknesses of the corresponding dielectric layers. Consequently, the applied gate voltage modifies the graphene chemical potential and conductivity, thereby dynamically tuning the effective impedance of each transmitarray unit cell while the dielectric contributions remain constant, [28], [29], [23]. It is worth mentioning that the capacitance of the hBN1 dielectric layer is not included in the equivalent surface impedance model because it remains constant and is independent of the applied bias voltage. Since this layer does not contribute to the voltage-dependent tuning mechanism, its electromagnetic effect is incorporated into the fixed dielectric loading of the multilayer structure rather than represented as an independent lumped circuit element. Consequently, only the voltage-dependent components are retained in the equivalent circuit formulation.
D. Equivalent Lumped RLC Circuit model of Proposed Hemispherical Transmitarray Unit Cell The complete voltage-controlled impedance model of the multilayer transmitarray unit cell, which incorporates the electromagnetic loading effects of the graphene sectors and surrounding dielectric layers, Fig. 1, is formulated as, Zcell (Vg ) = Zg (Vg ) + ZhBN2 + ZSiO2 +
1 jωCg
(59)
where Zg (Vg ), which is identical to Rs,(n,k),ξ (Vg ) in Eq.57 denotes the voltage-dependent graphene sheet impedance, while Z(hBN2 ) and Z(SiO2 ) represent the impedance contributions associated with the finite-thickness dielectric layers. The effective capacitance of Cef f,(n,k),ξ (Vg ) in Eq.57, which is equivalent to Cg in Eq.59, represents the direction-dependent gap capacitance arising from the fringing fields between neighboring graphene sectors along the ξ-polarized direction. By substituting the equivalent graphene RLC model and the
Fig. 6: Equivalent circuit model for the proposed hemispherical graphene transmitarray unit cell, including the dielectric capacitances of the SiO2 , CSiO2 , and upper hBN layer, ChBN2 . The graphene sheet impedance modeled by the series resistance of Rg and inductance of Lg , and the gap capacitance of Cg . The lower hBN layer is excluded from the circuit model because it remains invariant and acts as a constant parameter.
IV. L OCAL ABCD MODEL AND VOLTAGE -C ONTROLLED T RANSMISSION C OEFFICIENT The ABCD-matrix formulation provides a rigorous analytical framework for evaluating S21 of the proposed multilayer transmitarray antenna by accounting for the cascaded electromagnetic response, interlayer coupling, dielectric propagation, and tunable graphene surface impedance. The analytical framework
11
accurately captures the cascaded electromagnetic response, Accordingly, the local transmission coefficient of the graphenefinite-thickness effects of individual layers, and impedance loaded (n, k)-th unit cell is obtained from the ABCD parametransformations occurring throughout the multilayer structure. ters as, Furthermore, it accounts for the multiple internal reflections at S21,(n,k),ξ = the interfaces among the constituent material layers, including 2 the graphene sectors, hBN dielectric layers, gold FCC patches, B(n,k),ξ in A(n,k),ξ + Zout + Zin C(n,k),ξ + ZZout D(n,k),ξ and silicon substrate. For a dielectric layer under normal (71) incidence, the corresponding ABCD matrix can be generally expressed as, where Zin and Zout denote the characteristic impedances of the media surrounding the unit cell, respectively. cos(ki di ) jZi sin(ki di ) Pi = (63) −1 jZi sin(ki di ) cos(ki di ) V. A NALYTICAL AND P RACTICAL MAPPING FROM THE D ESIRED B EAM D IRECTION TO THE B IAS VOLTAGES OF where ki and Zi denote the propagation constant and characterG RAPHENE S ECTORS istic impedance of the i-th dielectric layer, respectively, given by, Assuming a horn antenna is positioned at the geometric cen√ ki = k0 εr,i (64) ter of the hemispherical lens, the incident electromagnetic wave propagates radially and illuminates the conformal metasurface η0 Zi = √ (65) aperture with an approximately uniform phase distribution. The εr,i objective is to synthesize a transmitted beam in the desired where η0 , and ϵr,i represent the free-space impedance, and direction, relative permittivity of the i-th dielectric layer, respectively. An ŝ0 = (sin θ0 cos ϕ0 , sin θ0 sin ϕ0 , cos θ0 ) (72) equation of Yd = Z1d represents the characteristic admittance of the dielectric layer. The polarization-dependent graphene sheet thereby supporting adaptive beam steering and continuous admittance of the (n, k)-th unit cell is obtained by inverting the tracking of moving targets. To achieve beam steering toward corresponding element of the anisotropic graphene impedance the desired direction ŝ0 under spherical-wave excitation from tensor, expressed as, the feedhorn, the required transmission phase response of −1 the multilayer stack (n, k)-th unit cell on the conformal Yg,(n,k),ξ = Zg,(n,k),ξ (66) hemispherical transmitarray aperture is formulated as follows, where Zg,(n,k),ξ represents the graphene surface impedance req horn ψ(n,k) = −k0 ŝ0 · r(n,k) − ∠Einc r(n,k) + C (73) component associated with the ξ-polarized electromagnetic response of the unit cell. Using the equivalent transmission- where k0 and r(n,k) denote the free-space wavenumber and the line representation, the graphene sheet is modeled as a shunt position vector of the (n, k)-th unit cell on the hemispherical aperture, respectively. A constant of C represents an arbitrary admittance and described by the following ABCD matrix, constant phase reference. We first establish a four-step analyti1 0 Sg,(n,k),ξ = (67) cal framework for determining the bias voltages of the graphene Yg,(n,k),ξ 1 sectors corresponding to desired beam directions of moving Similarly, the gold FSS layer is modeled as an equivalent targets. The practical realization of this framework is then shunt sheet admittance, with its corresponding ABCD matrix presented as an eight-step voltage-controlled beam-steering expressed as follows, synthesis procedure in Table III. Step 1: Analytical derivation of the required transmission 1 0 SAu,(n,k),ξ = (68) phase distribution: since the feedhorn phase center coincides YAu,(n,k),ξ 1 with the geometric center of our hemispherical lens, all unit where YAu,(n,k),ξ denotes the polarization-dependent sheet cells on the conformal aperture are approximately equidistant admittance of the gold FSS layer along the selected polarization from the feedhorn. Consequently, the incident phase distribution direction of ξ. across the hemispherical aperture is nearly uniform and can By incorporating the dielectric propagation layers, the be expressed as, complete transfer matrix of the (n, k)-th unit cell is obtained horn ∠Einc r(n,k) ≈ −k0 R (74) by cascading the individual layer matrices as, where R refers to the hemisphere radius. Since this propagation m(n,k),ξ = PSi PhBN1 Sg,(n,k),ξ PhBN2 SAu,(n,k),ξ (69) phase term is identical for all unit cells, it can be incorporated where PSi , PhBN1 , and PhBN2 represent the dielectric prop- into the constant phase reference of C, yielding, req agation matrices of the silicon and hBN layers, respectively, ψ(n,k) ≈ −k0 ŝ · r(n,k) + C (75) accounting for the phase accumulation and impedance transformation through the corresponding dielectric regions. The The dot product between the desired beam direction vector s0 resulting ABCD transfer matrix of the (n, k)-th unit cell can and the position vector r(n,k) of the (n, k)-th unit cell on the hemispherical aperture, which is expressed as, be written as, ŝ·r(n,k) = R sin Θ0 sin θn cos Φ0 − ϕ(n,k) + cos Θ0 cos θn A(n,k),ξ B(n,k),ξ m(n,k),ξ = (70) (76) C(n,k),ξ D(n,k),ξ
12
Substituting the Eq.76 into the required phase formulation in Eq.75 yields, req ψ(n,k) = −k0 R sin Θ0 sin θn cos Φ0 − ϕ(n,k) + cos Θ0 cos θn + C (77) Step 2: Required local transmission coefficient: assuming ideal phase-only modulation with a transmission magnitude approximately equal to unity, the required complex transmission coefficient of the stacked multilayer (n, k)-th unit cell is expressed as, jΦreq req T(n,k) = e (n,k) (78) where Φreq (n,k) denotes the required transmission phase obtained from the beam-steering phase synthesis. This general formulation can also account for amplitude variations by incorporating the desired transmission magnitude into the complex transmission coefficient, i.e., req req T(n,k) = T(n,k) e
jΦreq (n,k)
Eq.83 establishes the analytical mapping between the desired beam-steering phase profile and the required local surface impedance, which is subsequently synthesized through voltagecontrolled modulation of graphene conductivity. Step 4: Voltage-controlled impedance realization: the required or desired phase profile is then transformed into the corresponding local surface impedance Zsreq , which is physically synthesized through the voltage-controlled graphene conductivity σg (µc ). Finally, the required chemical potential is converted into the corresponding gate voltage Vg , enabling dynamic realization of the desired beam direction in the following path, req req req ψ(n,k) −→ T(n,k) −→ Zs,(n,k) −→ µc,(n,k) −→ Vg,(n,k) (84) Therefore, the required gate voltage of Vg,(n,k) corresponding to the (n, k)-th unit cell is determined by solving the following impedance-matching equation, h i −jΦreq (θ ,ϕ0 ) (n,k) 0 Zs,(n,k) Vg,(n,k) = 2Z0 e −1 (85)
(79)
where Zs,(n,k) (Vg,(n,k) ) represents the voltage-dependent surface impedance of the graphene-loaded unit cell. Substituting (n,k) where Treq ≤ 1. the graphene impedance expression given in Eq.57 into Eg.83 Step 3: Analytical mapping from required transmission coef- and separating the resulting complex equation into its real and ficient to graphene surface impedance: the required transmission imaginary components yields the following coupled equations, coefficient of each unit cell is analytically transformed into an 2 equivalent surface impedance, providing the essential bridge Re Zs,(n,k) (µc ) = 2Z0 1 − (cos Φreq (86) (n,k) ) between the synthesized electromagnetic response and the voltage-controlled physical implementation of the graphene Im Zs,(n,k) (µc ) = −2Z0 sin Φreq (87) (n,k) metasurface, target beam direction (or desired direction), req req req ψ(n,k) −→ T(n,k) −→ Zs,(n,k) (80) Accordingly, for the graphene-loaded RLC equivalent model illustrated in Fig. 6, the real and imaginary impedance-matching For a thin conductive metasurface represented by an equivalent equations are expanded into the following expressions, surface impedance sheet, the complex transmission coefficient Dξ′ 2 of the multilayer stack is directly determined by the effective 1 + ω 2 τ 2 = 2Z0 1 − (cos Φreq ) (88) (n,k) Gξ σ0 (µc ) surface impedance according to, 2Zs (81) 2Zs + Z0 where Zs represents the voltage-dependent surface impedance introduced by the graphene-loaded unit cell. This relation establishes the analytical link between the desired or required transmission response and the physical graphene conductivity modulation. If the stacked multilayer (n, k)-th unit cell is represented by an equivalent series surface impedance, the required equivalent surface impedance is obtained from the desired complex transmission coefficient as, ! 1 req Zs,(n,k) = 2Z0 (82) req − 1 T(n,k) T =
This expression establishes the direct mapping between the synthesized transmission response and the required local surface impedance, thereby defining the electromagnetic design target for the realization of voltage-controlled graphene unit cells. Substituting Eq.78 into Eq.82 yields the required local surface impedance for the (n, k)-th graphene-loaded unit cell, expressed as follows, −jΦreq req (n,k) − 1 Zs,(n,k) = 2Z0 e (83)
ωτ
1 Dξ′ − = −2Z0 sin Φreq (n,k) Gξ σ0 (µc ) ωCeff,ξ (µc )
(89)
where the graphene Drude conductivity term is given by, e2 µc τ (90) πℏ2 The effective capacitance accounting for both geometric capacitance and graphene quantum capacitance contributions is expressed as follows, !−1 πℏ2 vf2 1 Ceff,ξ (µc ) = + 2 (91) Cgeo,ξ 2e Dθ Dϕ |µc | σ0 (µc ) =
By solving the coupled nonlinear equations given in 86 and Eq.87, the required spatial distribution of graphene chemical potential of µc,(n,k) is analytically determined. The resulting chemical-potential distribution is then transformed into the corresponding gate-voltage map of Vg,(n,k) , establishing a direct electrical control mechanism for real-time programmable beam steering of the hemispherical transmitarray antenna. Because the resulting impedance-matching equations, Eqs.86 and 87, are transcendental due to the nonlinear dependence
13
of graphene conductivity, quantum capacitance, and effective surface impedance on the chemical potential, a closed-form analytical solution for Vg,(n,k) is generally not available. Therefore, a practical numerical optimization framework in Table III is developed to determine the voltage bias distribution of individual graphene sectors, enabling the physical realization of the synthesized electromagnetic phase profile for programmable beam steering. In this case, the required gate voltage for each graphene unit cell is obtained by solving the nonlinear inverse impedance-matching problem. The complex residual function is defined by, F (Vg ) = Zs (Vg ) − Zsreq = 0
TABLE III: Practical steps for mapping the desired beam direction to the bias voltages of the graphene sectors. Step
Flowchart Block
Mathematical and Operation Formulation
Resulting Output
Step 1
Beam Direction Specification
Target beam Define the desired beam-steering direction (θ0 , ϕ0 ) and the corresponding direction for unit direction vector from Eq. 72 performing steering
Step 2
Required Calculate the required phase distribution Φreq (n,k) Transmission Phase over all unit cells on the hemispherical Synthesis aperture from Eq.77
Step 3
Transmission Coefficient Synthesis
Convert the required phase profile into the desired complex transmission coefficient of Eq.78
Desired transmission response of each unit cell, req T(n,k)
Step 4
Required Surface Impedance Extraction
Map the transmission coefficient to the required local surface impedance using Eq. 83
Required surfaceimpedance distribution, req Zs,(n,k)
Step 5
Graphene Impedance– Voltage Mapping
Calculate the voltage-dependent Voltageimpedance relation of each dependent graphene-loaded unit cell, impedance Zs,(n,k) (Vg ), using, model • full-wave electromagnetic simulations • an analytical graphene RLC impedance model • a pre-computed impedance look-up table (LUT), Vg 7→ Zs
Step 6
Voltage Optimization for Each Unit Cell
For each graphene sector, determine the Voltage solution for each unit gate voltage by solving the cell impedance-matching condition,
(92)
the solution is computed iteratively using the Newton–Raphson method, [30], (i) F Vg (93) Vg(i+1) = Vg(i) − (i) F ′ Vg Since Vg is a real-valued control variable while Zs is complex, convergence is achieved by minimizing the complex impedance residual, 2 E(Vg ) = |Zs (Vg ) − Zsreq | → 0 (94) This procedure is repeated for every unit cell to determine the complete gate-voltage distribution required to realize the desired phase profile across the graphene transmitarray. VI. C ONFORMAL S PHERICAL A RRAY FACTOR FOR THE G RAPHENE - BASED T RANSMITARRAY H EMISPHERICAL A NTENNA
req
Zs,(n,k) (Vg ) − Zs,(s,k) → 0 (95)
Step 7
Numerical Root-Finding Algorithm
Optimize the gate voltage using Newton–Raphson or numerical minimization, Vg,(n,k) = arg min
Z
(V ) − Z
Minimum error impedance of Error(Vg ) = Zs,(n,k) (Vg )− req Zs,(n,k)
req
g s,(n,k) s,(n,k) 2 The array factor is a well-established concept in antenna Vg (96) theory that is rigorously formulated for passive antenna arrays. Integrating active components, such as amplifiers, transistors, Final Programmable Apply the obtained voltage distribution Real-time or dynamically tunable elements, extends this framework Out- Graphene Bias to electrically isolated graphene sectors, programmable n o beam steering beyond its conventional scope and complicates its theoretical put Map (1,1) (1,2) (N ,N ) and Vg , Vg , . . . , Vg n r formulation. For instance, active elements introduce spatially (97) moving-target tracking varying gain and phase responses that no longer satisfy the fundamental assumptions of the pattern multiplication theorem, which traditionally defines the total radiation pattern as the product of the isolated element pattern and the array factor. excitation weights of each graphene sectors in the following Accordingly, as a first step toward extending the classical formula, Nr X Nn array factor formulation developed for passive antenna arrays X jk0 s·r0 AF (Θ, Φ; p) = W0 e + Wn,k ejk0 s(Θ,Φ)·r(n,k) to our hemispherical transmitarray antenna, the dynamic n=1 k=1 tunability of the unit cells is incorporated into the analytical (98) framework. Although our hemispherical antenna does not where Nr and Nn represent the total number of concentric contain conventional active RF devices, it can be categorized rings and the number of unit cells in the n-th ring, respectively. as a semi-active transmitarray antenna because its radiation W0 denotes the apex excitation weight associated with the characteristics are governed by externally applied bias voltages. central unit cell of the hemispherical aperture. The complex These applied voltages dynamically modulate the surface excitation coefficient of each (n, k)-th unit cell is defined as conductivity of the graphene sections, thereby tuning the overall follows, performance. The most rigorous formulation is to merge the conventional free-space hemispherical array factor with the W(n,k) = ∆Sn Fhorn θn , ϕ(n,k) Fcell,(n,k) (Θ, Φ) S21,(n,k) graphene transmission coefficient of S21,(n,k) and the horn (99) illumination phase into a single conformal array factor. where Fhorn represents the incident horn illumination disHence, the conformal array factor of our hemispherical tribution over the hemispherical aperture. The individual transmit-array antenna is explicitly expanded in terms of unit-cell radiation pattern is denoted by Fcell,(n,k) . In the
14
initial optimization stage, the unit-cell radiation pattern is approximated as isotropic, i.e., Fcell,(n,k) (Θ, Φ) ≈ 1. This approximation significantly reduces the computational burden while retaining the dominant contributions of the hemispherical aperture geometry, feedhorn illumination, and voltagecontrolled transmission characteristics. S21,(n,k) represents the voltage-controlled complex transmission coefficient of the graphene-loaded unit cell, defining the amplitude and phase response of the transmitted electromagnetic field as a function of the applied bias voltage. Hence, the total far-field phase contribution from each unit cell is composed of three components: (1) the voltage-controlled transmission phase provided by the graphene sectors, (2) the geometrical propagation phase determined by the unit-cell position on the hemispherical aperture, and (3) the incident phase of the spherical wave generated by the feedhorn. Indeed, the total far-field phase is given by the superposition of the following three phase contributions, Φfar−field total,(n,k) =
(a)
Φtrans,(n,k) | {z }
graphene-controlled transmission phase
+
k0 ŝ · r(n,k) | {z }
geometrical propagation phase
horn + ∠Einc r(n,k) | {z }
horn illumination phase
(100) Hence, the phase contribution in the array factor in Eq.98 can be explicitly expressed as, desired
ejk0 ŝ(Θ,Φ)·r(n,k) = e−jΦ(n,k) × exp jk0 R sin θ sin θn cos ϕ − ϕ(n,k) + cos θ cos θn (101) desired
Specifically, the component of e−jΦ(n,k) represents the synthesized transmission phase programmed to steer the radiated beam toward the desired direction, while the second exponential term describes the free-space propagation phase determined by the location of the (n, k)-th unit cell on the hemispherical aperture and the observation direction (θ, ϕ). Their product ensures coherent superposition of the radiated fields, thereby forming a high-gain beam in the desired steering direction while suppressing radiation in undesired directions. VII. S IMULATION R ESULTS AND P ERFORMANCE E VALUATION
(b)
Fig. 7: Phase variations of S21 for the proposed hemispherical graphene-based transmitarray antenna: (a) frequency-dependent phase response and (b) chemical-potential-dependent phase response. For each case, a uniform chemical potential is applied simultaneously to all 121 graphene elements to evaluate the overall tunable transmission-phase characteristics of the metasurface.
The proposed graphene-based hemispherical transmitarray antenna in Fig.1 is validated using the full-wave electromagnetic solver CST microwave Studio. In the CST Studio Suite environment, the hemispherical transmitarray antenna unit cells and highlight the capability of the proposed transis excited by a circular feed horn located at the center mitarray architecture for dynamic electromagnetic wavefront of the hemispherical lens, providing a realistic illumination control. Simulation results in Figs. 9 and 10 have verified with nonuniform amplitude and phase distributions across the that the proposed hemispherical transmitarray antenna enables metasurface aperture. The simulation results obtained from wide-angle beam steering, covering approximately 90◦ in CST microwave Studio were exported to MATLAB for post- the elevation direction and a full 360◦ range in the azimuth processing, analysis, and graphical visualization. The phase direction. Such comprehensive spatial coverage highlights the variations of S21 with respect to frequency and graphene advantage of the hemispherical geometry and voltage-controlled chemical potential are presented in Fig. 8. These results graphene reconfiguration over conventional planar metasurface demonstrate the tunable phase response of the graphene-based antennas reported in the literature in Table II.
15
(a)
(b)
Fig. 8: Distribution of the transmission coefficient of S21 over our proposed hemispherical graphene-based transmitarray antenna of Fig.1. In this case, a uniform graphene chemical potential of µc = 0.25 eV is assigned to all 121 graphene sectors. The hemispherical transmitarray antenna is illuminated by a circular feedhorn with an aperture size of 5 mm in the CST Studio Suite environment to excite the curved lens. The obtained results S21 are subsequently exported to MATLAB for post-processing and visualization of the transmission characteristics across our overall hemispherical transmitarray antenna: (a) transmission-amplitude, |S21 |, distribution and (b) transmission-phase, ∠S21 , distribution.
Fig. 10: Three-dimensional distribution of the normalized transmitted field over the proposed hemispherical transmitarray aperture for evaluating beam-scanning performance along the θ and ϕ angular directions: (a) θ = 0◦ , ϕ = 0◦ , (b) θ = 45◦ , ϕ = 45◦ , (c) θ = 75◦ , ϕ = 250◦ , and (d) θ = 90◦ , ϕ = 300◦ .
across the hemispherical aperture, ηcell,(n,k) ≈ S21,(n,k)
ηlens ≈
VIII. L ENS T RANSMISSION E FFICIENCY For the proposed hemispherical transmitarray lens, consistent with Fig. 1, the local transmission efficiency is governed by the transmission coefficient of S21,(n,k) for the (n, k)-th unit cell, which defines the local transmission amplitude and phase
(102)
This quantity represents the fraction of incident power locally transmitted by the cell, under the assumptions of the equivalentcircuit or full-wave unit-cell model. It is worth noting that 2 S21,(n,k) quantifies only the local transmission efficiency of the (n, k)-th unit cell, without accounting for horn-feed mismatch, aperture spillover, overall radiation efficiency, beampointing losses, or sidelobe power distribution. Therefore, 2 S21,(n,k) should be interpreted as the local transmission efficiency of the (n, k)-th transmitarray unit cell. For our proposed hemispherical transmitarray lens with 121 gold patches, the overall transmission efficiency is expended in terms of the horn-illumination-weighted average of the local transmitted power over the entire aperture given by, m X
Fig. 9: Three-dimensional distribution of the normalized transmitted field over the proposed hemispherical transmitarray aperture for evaluating beam-scanning performance along the θ and ϕ angular directions: (a) θ = 0◦ , ϕ = 0◦ , (b) θ = 45◦ , ϕ = 0◦ , (c) θ = 70◦ , ϕ = 120◦ , and (d) θ = 60◦ , ϕ = 60◦ .
2
∆S(n,k) Fhorn,(n,k)
2
S21,(n,k)
(n,k)=1 m X
2
(103) ∆S(n,k) Fhorn,(n,k)
2
(n,k)=1
where m denotes the total number of graphene-based unit cells. The complex incident field (or illumination amplitude) of the horn at the (n, k)-th unit cell is expressed by Fhorn,(n,k) . The complex transmission coefficient, S21,(n,k) , is associated with the corresponding graphene–hBN–gold unit cell. The 2 illumination factor of ∆S(n,k) Fhorn,(n,k) characterizes the nonuniform horn illumination over the hemispherical aperture and represents the power intercepted by the (n, k)-th unit cell. The metric of ηlens quantifies how efficiently our metasurface lens transmits the electromagnetic power incident from the circular feedhorn through the hemispherical aperture, providing an estimate of the average transmitted power through the
16
TABLE IV: Comparison of graphene-based antenna technologies in terms of antenna efficiency. Refs.
Frequency Range
Antenna Type
Efficiency
Reconfiguration Mechanism
[31]
5.8 GHz, 1–2 THz 0.276– 0.711 THz
[33]
THz band (2.32 THz bandwidth) 889–950 MHz
MW: 16–66%; THz: 26–69% Not reported (ECC < 0.01, DG ≈ 10 dB) 21.5%
DC-biased graphene conductivity tuning
[32]
Planar microstrip patch with parasitic graphene Tree-shaped graphene-based wideband printed MIMO antenna Graphene dipole antenna
Printed graphene dipole RFID antenna
40%; 2.18 dBi gain
Hemispherical graphene transmitarray with FCC patches
Achieves values beyond the 68%–80% range
No reconfiguration (passive graphene conductor) Voltage-controlled graphene beam steering
[34]
This work
200–300 GHz
Chemical-potential tuning
DC-biased graphene conductivity tuning
lens under horn illumination. Therefore, it is highly effective for evaluating and comparing different graphene bias states, chemical potential distributions, and different geometrical designs. The CST Simulation results have verified that the overall antenna efficiency, Table IV, of the proposed hemispherical transmit array antenna ranges from 68% to 80% for different graphene voltage excitation states. Each bias configuration results in a specific electromagnetic response and corresponding efficiency level. The reported values conservatively include a 10% efficiency reduction to account for practical imperfections.The proposed metasurface transmitarray antenna demonstrates a substantial efficiency enhancement compared to typical available graphene-based antenna technologies in Table IV. IX. C ONCLUSION To conclude, this work presents the design, analytical modeling, and electromagnetic characterization of a voltagecontrolled hemispherical transmitarray antenna operating at 250 GHz, with the objective of enabling wide-range beam steering for real-time moving-target tracking applications. We have implemented stacked layers of FCC gold patches integrated with tunable graphene sectors, isolated by hBN dielectric layers, to achieve dynamic control of the transmitted electromagnetic wave and subsequent beam steering through spatially localized chemical-potential modulation and surfaceimpedance tuning by the following procedure, µc → σg (µc ) → Zs (µc ) → transmission phase of ϕt → beam-steering direction of (θ, ϕ) Independent and distinct chemical potentials ranging from 0.1 eV to 1 eV are assigned to the 121 graphene elements to provide flexible spatial phase manipulation and dynamic beam steering. The hemispherical transmit array antenna has exhibited a significant enhancement in its scanning capability, achieving approximately between −90◦ and +90◦ elevation coverage and a full azimuthal scanning range of 360◦ , and thereby
overcoming the angular limitations commonly associated with conventional planar antennas, as discussed in Table II. Our proposed metasurface transmitarray antenna also exhibits an antenna efficiency ranging from 68% to 80%, incorporating a conservative 10% degradation from the ideal theoretical performance to account for fabrication imperfections. A drastic difference in antenna efficiency is observed between available graphene-based antenna technologies, as summarized in Table IV, and the proposed metasurface transmitarray antenna. Furthermore, the integration of the proposed antenna with an AMD Xilinx Zynq UltraScale+ RFSoC ZCU670 platform validates the feasibility of real-time bias-matrix control and enables ultrafast adaptive beam steering and target tracking, offering a significant advancement over existing beam-steering technologies in Table I. Indeed, the synergistic combination of hemispherical geometry, voltage-controlled graphene reconfigurability, and scalable FPGA-based bias control provides a promising platform for adaptive beam steering and dynamic target tracking in future 6G wireless systems.
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