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Abstract
Compared with traditional electrical logic gates, optical or terahertz (THz) computing logic gates have faster computing speeds and lower power consumption, and can better meet the huge data computing needs. However, there are limitations inherent in existing optical logic gates, such as single input/output channels and susceptibility to interference. Here, we proposed a new approach utilizing polarization-sensitive graphene-vanadium dioxide metasurface THz logic gates. Benefitting from two actively tunable materials, the proposed controlled-NOT logic gate(CNOT LG) enables versatile functionality through a dual-parameter control system. This system allows for the realization of multiple output states under diverse polarized illuminating conditions, aligning with the expected input-output logic relationship of the CNOT LG. Furthermore, to demonstrate the robustness of the designed THz CNOT LG metasurface, we designed an imaging array harnessing the dynamic control capabilities of tunable meta-atoms, facilitating clear near-field imaging. This research is promising for advancing CNOT LG applications in the THz spectrum. It has potential applications in telecommunications, sensing, and imaging.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the age of big data, there’s a growing need to gather and process massive volumes of data, which places greater demands on the integration and processing speed of data computing devices. Traditional electronic computing equipment suffers from the low computing speed and efficiency induced by inherent ohmic losses and impedance matching issues [1–4]. Hence, the need for superior solutions for logic calculations are emerging. Optical and THz logic gates could significantly reduce the power consumption by circumventing the process of converting optical signals into electrical signals [5]. Moreover, benefiting from the high-speed performance of optical and THz devices, they emerge as the most promising alternative to electronic computing equipment [6–8]. Besides, traditional optical logic computing equipment still encounters several limitations, including stringent requirements on light source intensity and phase difference [8], poor anti-interference capabilities [9,10], complexity of equipment, etc.
To address the aforementioned limitations, many reports have investigated the application of metasurfaces for achieving optical or THz logic gate functionalities [11–13]. Metasurfaces are a novel type of artificial electromagnetic materials known for their sub-wavelength dimensions. Through deliberate design of unit structures and careful selection of materials, metasurfaces enable the realization of functionalities beyond the reach of conventional natural materials [14–18]. They offer solutions to the constraints of traditional optical devices, which tend to exhibit bulkiness and are characterized by single functionalities. Metasurfaces have found widespread applications in chiral manipulation [19], wavefront control [20,21], holographic imaging [22], electromagnetic absorber [23], vortex beam generation [24–26], etc. Moreover, they offer structural integration, ultra-low power consumption, and high operational efficiency, making them highly promising for implementing logic gates.
Furthermore, actively tunable materials like graphene, vanadium dioxide(VO$_2$), Ge$_2$Sb$_2$Te$_5$ (GST) and Bi$_2$-x$_2$bxTe$_3$(BST) exhibit diverse crystal states through electrical or chemical doping or alterations in external temperature. This capability allows for the modulation of their conductivity [27]. Integration of these materials into metasurfaces enables the realization of multifunctionality across different states [28]. This inherent adaptability renders them advantageous for implementing logic gates. While some relevant researches have been reported [29,30], current investigations in optical logic gates remain primarily focused on basic logic operations. Further research is imperative to meet the need for increasingly complex logic operations.
In this work, we introduce two metasurfaces for THz CNOT LGs, leveraging active tunable materials suitable for both linearly and circularly polarized waves. Compared to the higher phase transition temperature of GST and the more difficult operability of BST in our device structure [31,32], we chose two tunable materials, graphene and VO$_2$ to be arranged in three layers of split ring resonators (SRRs). We extract key parameters governing the tunable performance of these materials, namely the Fermi level of graphene and the electrical conductivity of vanadium dioxide. Through this dual-parameter control system, we achieved CNOT LG. Our designed metasurfaces exhibit distinct focusing effects under varying polarized plane wave conditions, with input and output states aligning with the logical functions of CNOT LG. Moreover, to underscore the robustness of the tunable CNOT LG, we demonstrate clear near-field imaging using an array composed of meta-atoms in different switching states. Our work underscores the great application of CNOT LGs within the THz band.
2. Design and methods
Figure 1 illustrates the operational principle of the CNOT LG metasurface. Two distinct metasurfaces were devised to accommodate linearly polarized (LP) waves and circularly polarized (CP) waves, respectively. Notably, the different design principles are adopted in two scenarios due to the unique requirements of LP and CP.
Fig. 1. Structure of metasurface for the proposed THz CNOT LG. (a) Diagram of metasurface structure under LP illumination. (b)Diagram of metasurface structure under CP illumination.
CNOT LG is a dual-qubit logic gate, where the high-order bit serves as the control bit, and the low-order bit functions as the target bit. Denoting the input as |ab> and the output as |AB>, the relationship between input and output can be expressed as Table 1.
Table 1. Input-output relationship of CNOT LG
Table 1 shows the definition of CNOT LG. Within the CNOT LG, the high-order bits remain unaltered. Specifically, when the high-order bits are 0, the low-order bits remain unchanged. Conversely, when the high-order bits are 1, the low-order bits undergo inversion.
2.1 CNOT LG for LP THz waves
Figure 1(a) depicts the metasurface structure of a THz CNOT LG working under LP THz wave illumination. The configuration comprises three layers of donut-shaped split ring resonators (SRRs) separated by a polyimide(PI) dielectric layer with the thickness of 10 $\mathrm{\mu}$m. To establish a dual parameters control system, we designed a SRR layer that employs vanadium dioxide (VO$_2$) sandwiched between two identical SRR arrays adopted graphene which possess distinct structural parameters of VO$_2$. This design exhibits great potential in independent control and modulation-specific working layers.
The metasurface array presented in Fig. 1(a) comprises 80$\times$80 periodic meta-atoms with a side length of 100 $\mathrm{\mu}$m. The phase modulation of LP incident THz wave is achieved through the control of the propagation phase, which necessitates varying structural parameters of each SRR. Upon irradiation by LP THz, the relationship between the transmitted wave and the incident wave on the metasurface can be described by the following Jones matrix in the Cartesian coordinate system, as shown in Eq. (1).
As shown in Eq. (2), to attain the focusing effect on a particular transmitted component of the wave, the phase distribution of said component exhibited by the metasurface must adhere to:
where $f_i,(i\in \{1,2\})$ is the focal length of the designed metalens, is the wavelength of the incident wave. $(x,y)$ is the coordinate of each meta-atom with the center as the origin in the Cartesian coordinate system, and $\varphi (x,y)$ is the phase profile distribution for generating metalenses, respectively. As illustrated in Fig. 1(a), the diverse structural parameters of individual atoms within the metasurface lead to varying phase delays across each transmission component. To realize the focal function upon the illumination of LP, eight meta-atoms are meticulously arranged with a phase gradient, enabling them to cover the full 2$\pi$ range within the cross-polarized component.In the proposed metasurfaces, the upper and lower arrays of SRRs based on graphene create the metalens with a focal length of $f_1$ = 6.5 mm. Considering other scenarios, when different polarized wave illuminates the metasurface, metalenses with different focal lengths need to be designed. Therefore, it is necessary to induce another adjustable parameter to form a dual-parameter system establishing metalens with four input and four output functions. The middle layer SRR array adopted VO$_2$ material for the phase change characteristic, and achieved a metalens with the focal length of $f_2$ = 4.5 mm.
Among the actively tunable materials candidates, graphene stands out for the distinctive optoelectronic characteristics. A critical parameter influencing the performance of the designed metasurface is the surface conductivity in the THz band, which can be evaluated using the Kubo formula: $\sigma _g = \sigma _{inter} + \sigma _{intra}$. Consequently, the overall surface conductivity of graphene is comprised of two components: inter-band transition and intra-band transition. This conductivity can be approximated as Eq. (3) [33,34].
The Fermi level of graphene can be effectively manipulated by varying the carrier concentration. Commonly employed techniques for achieving this include applying an external electric field and adjusting electrical or chemical doping [38]. At varying Fermi level, graphene exhibits metallic and insulating states, which enables our metalens to seamlessly transition between operational and non-operational states. On the other hand, VO$_2$ is a phase-change metal oxide, wherein its conductivity can be tuned via laser irradiation. At an ambient temperature of 300 K, VO$_2$ remains in a transparent insulating state without laser exposure with a dielectric constant of 9 S/m in the THz band. Sufficiently intense laser irradiation induces VO$_2$ transiting from insulating to metallic state. Post-transition, the electrical conductivity of the fully metallic VO$_2$ can reach 200000 S/m [39], aligning with the Drude formula [39] as depicted in Eq. (7):
In the device production process, graphene films is produced by chemical vapor deposition. Graphene SRRs can be patterned with 20-kV electron beam lithography using AR-N 7520 (ALLRESIST) as a negative electron beam resist and etched in oxygen plasma after development [44]. Furthermore, VO$_2$ thin films can be deposited along c-axis on single crystalline sapphire substrates by pulsed laser deposition (PLD) [45]. The negative photoresist (NR7-3000PY) can be spin coated over the VO$_2$ films and exposed by a UV light through a mask to pattern the SRR structures [46].
Figure 2(a) depicts a schematic of the graphene SRR unit in Fig. 1(a). The upper and lower layer shares the same unit, collectively forming a metalens with a focal length of 6.5 mm. Each unit features a periodic side length (L) of 100 $\mathrm{\mu}$m, a width (w) of 10 $\mathrm{\mu}$m, and a radius (r) of 45 $\mathrm{\mu}$m.The thickness of a single layer of graphene is about 0.34 nm. During the modeling and simulation process, we set the thickness of graphene to 1 nm [47], and the relaxation time to 1 ps [37]. The temperature of graphene is adjusted accordingly to match the required operating temperature of VO$_2$. Utilizing numerical simulation software (CST 2019, Microwave Studio), the splitting angle of the rings is varied, yielding the structural parameters illustrated in Fig. 2(a). The four units depicted in Fig. 2(a) possess a splitting direction ($\varphi$1) of 135$^{\circ }$ and are labeled "2", "3", "4", "5". As shown in Fig. 2(b), the phase delay (Pyx) of the linearly cross-polarized channel for these four units covers 180 degrees. Rotating these units 90$^{\circ }$($\varphi _1=45^\circ$) clockwise generates another set of four units with ID of "6", "7", "0", "1". Collectively, the eight units fully encompass the phase range of 2$\pi$. Additionally, Fig. 2(b) presents the simulated amplitude of the linearly cross-polarizated channel under different Fermi level of bilayer graphene. Setting $E_f$ = 0.01 eV, the transmitted amplitude for cross-polarization channel (tyx) at 1 THz remains below 0.1, indicating only a minute portion of the incident LP THz wave is transformed into cross-polarized components by the units. Conversely, at the Fermi level of 1 eV, the amplitude of the cross-polarization component from the eight units consistently maintains near 0.45, signifying efficient conversion of incident LP waves into orthogonal polarized waves. To deepen our understanding of the tunable physical phenomenon, we scrutinize the electric field distribution on the SRR surface under the condition where the Fermi level of graphene is set to 1 eV. As shown in Fig. 2(c), when the terahertz wave is incident on the surface of a SRR unit, the free electrons on the surface of graphene in a metallic state are excited to produce a collective resonance phenomenon, generating an electromagnetic surface wave that propagates along the metal-medium interface, namely surface plasmon polaritons(SPPs) [48–50]. As a result, the surface electric field of the unit is enhanced and augmenting polarization conversion capabilities. Conversely, Fig. 2(d) depicts a much weaker electric field distribution on the SRR surface at the Fermi level of 0.01 eV compared to that of 1 eV, attributable to graphene’s low conductivity, which restrains significant electromagnetic resonance effects.
Fig. 2. Design of units under LP working conditions. (a) Diagram of the SRR structure and size of the graphene layer. L = 100 $\mathrm{\mu}$m, w = 10 $\mathrm{\mu}$m, r = 45 $\mathrm{\mu}$m, the units numbered "6","7","0","1" is obtained by rotating the units numbered "2","3","4","5" by 90 degrees without changing the structure parameters. (b) Amplitude and phase retardation (red line) of the lineariy cross-polarized components of different units (numbered 0-7) when the Fermi level of graphene is at 1 eV (blue line) and 0.01 eV (grey line). (c) Surface electric field distribution setting $E_f$ = 1 eV. (d) Surface electric field distribution setting $E_f$ = 0.01 eV. (e) Diagram of the SRR structure and size of the VO$_2$ layer. (f) Amplitude and phase retardation (red line) of linely cross-polarized components of different units (numbered 0-7) when the conductivity of VO$_2$ is $2 \times 10^5$ S/m (blue line) and 10 S/m (gray line). (g) Surface electric field distribution when the conductivity of VO$_2$ is $2 \times 10^5$ S/m. (h) Surface electric field distribution of VO$_2$ when the conductivity is 10 S/m.
Figure 2(e) presents the schematic of the VO$_2$ SRR within CNOT LG metasurface under the LP terahertz wave. During the simulation we set the thickness of VO$_2$ to 200 nm [42,51]. Analogous to Fig. 2(a)-(d), the splitting angle and direction of the SRRs are adjusted to identify eight units covering 0-2${\pi }$ of phase shift. As illustrated in Fig. 2(f), setting $\sigma = 2 \times 10^5$ S/m, the linearly cross-polarized transmittance of the eight units at 1 THz remains consistently 0.45.Figure 2(g) displays the electric field distribution on the SRR surface under this phase change material state, revealing a robust surface electric field owing to the resonance effect induced by VO$_2$’s metallic conductivity, thereby enhancing the polarized conversion capability of the designed metasurfaces. Conversely, when the conductivity of VO$_2$ is setting to $\sigma = 10$ S/m, the linearly cross-polarized transmittance approaches zero. As evident by Fig. 2(h), the electric field intensity on the SRR surface nearly diminished. In this case, our metasurface exhibits negligible influence on the linearly polarization incident wave and lacks polarized conversion capability.
Obviously, employing different operational conditions can lead graphene and VO$_2$ be independently manipulated into metallic or insulating states. Consequently, distinct effective working layers will dictate the switching states of the two metalenses, yielding four distinct focusing effects of CNOT LGs. As depicted in Fig. 3, x-polarized wave is denoted as the control bit "0", while the y-polarized wave is labeled as the control bit "1". With the conductivity of VO$_2$ set at 10 S/m, the lower Fermi level of graphene corresponds to target bit "0", and the higher Fermi level is target bit "1". Conversely, when the conductivity of VO$_2$ is set as $2 \times 10^5$ S/m, the lower Fermi level of graphene is recorded as target bit "1", and the higher Fermi level is denoted as target bit "0". This definition rule is completely known and controllable in the actual operation process [12,52,53]. At the output end, the presence of focus is denoted as "1", while the absences indicated as "0". Additionally, the focus state of the shorter focal length is designated as the high bit, while the longer one is assigned the low bit.
Figure 3(a) and Fig. 3(b) illustrate the working principles of CNOT LG under x-polarized terahertz wave illumination. As depicted in Fig. 3(a), setting $E_f$ = 0.01 eV and $\sigma = 10$ S/m, none of the layers are working, resulting in the absence of focus. Conversely, as shown in Fig. 3(b), setting $E_f$ = 1 eV and $\sigma = 10$ S/m, the graphene layer is active while the VO$_2$ layer remains inactive. Consequently, a focus is achieved at the distance of 6.5 mm with x-polarized wave illumination. On the other hand, for the dual-parameters control system, the conductivity of VO$_2$ is another key tunable parameter. The metallic VO$_2$ units work under y-polarized wave illumination to form the meta-lens with a focal length of 4.5 $\mathrm{\mu}$m. Figure 3(c) and Fig. 3(d) depict the working principles under y-polarized wave illumination. In Fig. 3(c), when the Fermi level of graphene is 1 eV and the conductivity of VO$_2$ is $2 \times 10^5$ S/m, all layers are active. Consequently, the y-polarized terahertz wave incident on the metasurface results in the generation of two focal points at 4.5 mm and 6.5 mm, respectively. On the other hand, as depicted in Fig. 3(d), only the VO$_2$ layer is working under the condition of $E_f$ = 0.01 eV and $\sigma = 2 \times 10^5$ S/m. In this case, there is a focus solely at 4.5 mm distance under y-polarized wave illumination. The four working states depicted in Fig. 3 align perfectly with the logical relationship of CNOT LG. Our THz CNOT LG metasurface is realized by leveraging the tunable characteristics of graphene and VO$_2$.
Fig. 3. Working principle of THz CNOT LG metasurface for LP THz wave . With different control states and incident conditions, the metasurface shows (a) no focus "00", (b) a long focal length focus "01", (c) two focus points "11", (d) a short focus Focal length of focus "10".
2.2 CNOT LG for CP THz waves
Different from the case of LP THz wave illumination, the phase control of CP waves can be accomplished via the geometric phase. The electric field amplitude of right circularly polarized (RCP) wave is denoted as $E_{R}$, and left circularly polarized (LCP) wave is denoted as $E_L$. In the circular base, the electric field amplitude vector of RCP is represented as $\boldsymbol {E} _{RCP}=[E_{R} \ 0]^T$, and LCP is denoted as $\boldsymbol {E}_{LCP}=[0 \ E_{L}]^T$. The transformation matrix $\boldsymbol {\varLambda }$ from the circular base to the Cartesian base is as shown in Eq. (9). Therefore, in the Cartesian base, the electric field amplitude vector of RCP is expressed as $\widehat {\boldsymbol {E}}_{RCP}=\sqrt {2}/2[E_R \ iE_R]^T$, and LCP is denoted as $\widehat {\boldsymbol {E}}_{LCP}=\sqrt {2}/2[E_L \ -iE_L]^T$. Generally, $\boldsymbol {J}$ is the Jones matrix of meta-atom. After the meta-atom is rotated with an angle $\theta$, the rotated Jones matrix is represented as $\boldsymbol {J}^{\boldsymbol {R}}=\boldsymbol {R}\left ( \theta \right ) \boldsymbol {JR}\left ( -\theta \right ) =\boldsymbol {R}\left ( 2\theta \right ) \boldsymbol {J}$. Here, the rotation matrix $\boldsymbol {R} (\theta )=\left ( \begin {matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end {matrix} \right )$, then:
Combined with Eq. (10) and Eq. (11), the transmitted wave [54] when RCP wave is incident can be expressed as shown in Eq. (12).
The transmitted wave when illuminated by LCP can be expressed as shown in Eq. (13).
Obviously, when the meta-atom is rotated by an angle of $\theta$, the orthogonal polarization component transmitted under RCP or LCP illumination will be appended with a rotation angle phase of 2$\theta$ or -2$\theta$. Therefore, as illustrated in Fig. 1(b), the structural parameters of each meta-atom in the designed metasurface for CP wave remain consistent, with only a rotational operation capable of achieving a 2$\pi$ range of the cross-polarized channel.
Next, as shown in Fig. 4, we analyze the THz CNOT LG metasurface for CP wave. Similarly, the periodic side length L of the units in the metasurface is 100 $\mathrm{\mu}$m. The graphene SRR unit for the RCP wave illumination is shown in Fig. 4(a), where $r_3$ = 34.4 $\mathrm{\mu}$m, $w _3$ = 24.8 $\mathrm{\mu}$m, $\alpha _3$ = 60$^{\circ }$. Rotating the unit in accordance with the principles of geometric phase allows for the regulation of the phase of the cross-polarized component. As shown in Fig. 4(b), eight units with bilayer graphene are obtained by rotating different angles $\varphi _3$ so that the phase of the cross-polarized component covers the range of 0-2$\pi$, meeting the conditions for focusing. Additionally, setting $E_f$ = 1 eV, the cross-polarized conversion efficiency of the eight units at 1 THz under normal incidence of RCP are consistently maintained near 0.5. Figure 4(c) depicts the surface electric field distribution at this state. The conductivity of graphene resonates with the incident wave, enabling it to excite the SPPs principle and granting it the ability for cross-polarization conversion. However, graphene loses its conductivity under the condition of $E_f$ = 0.01 eV. Consequently, the conversion efficiency falls below 0.1, rendering it essentially devoid of conversion capability. Correspondingly, the surface electric field shown in Fig. 4(d) is also notably weak.
Fig. 4. Design of units under CP working conditions. (a) Diagram of the SRR structure and size of the graphene layer. L = 100 $\mathrm{\mu}$m, $w_3$= 24.8 $\mathrm{\mu}$m, $r_3$ = 34.4 $\mathrm{\mu}$m. (b) The amplitude and phase retardation (red line) of the circular cross-polarization component of different meta-atoms (numbered 0-7) obtained by rotation under the conditions of $E_f$ = 1 eV (blue line) and $E_f$ = 0.01 eV (grey line). (c) Surface electric field distribution of graphene under normal incidence of RCP when the Fermi level is 1 eV. (d) Surface electric field distribution of graphene under normal incidence of RCP when the Fermi level is 0.01 eV. (e) Diagram of the SRR structure and size of the VO$_2$ layer. L = 100 $\mathrm{\mu}$m, $w_4$= 25 $\mathrm{\mu}$m, $r_4$ = 40 $\mathrm{\mu}$m. (f) The amplitude and phase delay (red line) of the circular cross-polarization components of different meta-atoms (numbered 0-7) obtained by rotation when the conductivity of VO$_2$ is $2 \times 10^5$ S/m (blue line) and 10 S/m (gray line). (g) Surface electric field distribution of VO$_2$ under reverse incidence of LCP when the conductivity is $2 \times 10^5$ S/m. (h) Surface electric field distribution of VO$_2$ under reverse incidence of LCP when the conductivity is 10 S/m.
On the other hand, Fig. 4(e) showcases the VO$_2$ SRR suitable for irradiation with LCP waves, featuring r$_4$ = 40 $\mathrm{\mu}$m, $w _4$ = 25 $\mathrm{\mu}$m, $\alpha _4$ = 60$^{\circ }$, $\varphi _4$ = 45$^{\circ }$. By rotating this unit, the phases of eight meta-atoms can satisfy the focusing requirement. Figure 4(f) presents the cross-polarized conversion efficiency under LCP back incidence. When VO$_2$ is in the metallic state, the cross-polarization conversion efficiency remains stable around 0.35. Conversely, the conversion efficiency is essentially zero when VO$_2$ is insulating. Figure 4(g) and Fig.4(h) depict respectively the surface electric field when VO$_2$ is in a metallic state and an insulating state. This phenomenon resembles that shown in Fig. 4(c) and Fig. 4(d), fully demonstrating the excellent tunable characteristics of graphene and VO$_2$.
Similar to the LP CNOT LG metasurface depicted in Fig. 1(a), the metasurface for CP wave in Fig.1(b) also consists of three-layer annular SRRs, forming two lenses with different focal lengths. The upper and lower SRR layers are fabricated from graphene material, while the middle layer SRR is composed of VO$_2$ material. The adjustable Fermi level of graphene and the conductivity of VO$_2$ are two pivotal parameters that constitute a dual-parameter control system, which can determine the working performance of THz CNOT LG. As shown in Fig. 5, RCP is designated as the high-order control bit "0", while LCP is labeled as the high-order control bit "1". Other labeling conventions remain consistent with Fig. 3.
Fig. 5. Working principle of THz CNOT LG metasurface for CP THz wave. Under different control conditions and incident states, the metasurface exhibits (a) no focus "00", (b) a focus with the long focal length "01", (c) two focus points "11", (d) a focus with the short focal length "10".
Considering the situation when RCP is incident on the metasurface. As depicted in Fig. 5(a), none of the layers are working under the condition of $E_f=0.01$ eV and $\sigma =10$ S/m. Consequently, RCP will not result in the formation of a focus after normal incidence. However, as illustrated in Fig. 5(b), with $E_f = 1$ eV and $\sigma =10$ S/m, the graphene layer becomes working while the VO$_2$ layer remains non-working. Consequently, RCP will generate a focus at 6.5 mm after normal incidence. In additon, LCP is marked as "1" on the high bit of the input terminal in the designed THz CNOT LG. The LCP illumination is also a critical working condition for the designed metasurface. Since the metasurface generated based on the geometric phase principle has conjugate properties under the illumination of different CP waves, the LCP will be scattered under normal incidence and focused under backward incidence. Therefore, as shown in Fig. 5(c), all layers work under the condition of $E_f=1$ eV and $\sigma =2\times 10^5$ S/m. The VO$_2$ layer is designed as a meta-lens that can focus at 4.5 mm at the transmission end under LCP back incidence. Therefore, two focus points can be generated at 4.5 mm and 6.5 mm. As shown in Fig. 5(d), only the VO$_2$ layer works under the conditions of $E_f = 0.01$ eV and $\sigma =2 \times 10^5$ S/m. In this case, only one focus point is generated at 4.5 mm. The designed metasurface for CP wave is also fully capable of realizing the logic functions of CNOT LG.
3. Result and discussion
3.1 CNOT LG for LP THz wave
The eight units shown in Fig. 2 are arranged according to the phase profile described by Eq. (2), resulting in an 80$\times$80 unit array as depicted in Fig. 1(a). The focal length of the metalens composed of the middle VO$_2$ SRR array is $f_A$ = 4.5 mm, while the focal length of the metalens composed of the upper and lower graphene SRR arrays is $f_B$ = 6.5 mm. Figure 6 shows the simulation results of the metasurface under four conditions, with their working conditions corresponding to Fig. 3(a-d), respectively.
Fig. 6. Far-field simulation results and focus intensity distribution of xoz plane and xoy plane under different conditions : (a) $\sigma$ (VO$_2$) = 10 S/m, $E_f$ (Graphene) = 0.01 eV, x-LP normal incidence. (b) $\sigma$(VO$_2$) = 10 S/m, $E_f$(Graphene) = 1 eV, x-LP normal incidence. (c) $\sigma$(VO$_2$) = $2\times 10^5$ S/m, $E_f$(Graphene) = 1 eV, y-LP normal incidence. (d) $\sigma$(VO$_2$) = $2\times 10^5$ S/m, $E_f$(Graphene) = 0.01 eV, y-LP normal incidence. Among them, plane A is the xoy plane with z = 4.5 mm, and plane B is the xoy plane with z = 6.5 mm. When there is focus on the focal plane, the corresponding bit is "1", and when there is no focus, the corresponding bit is "0".
Figure 6(a) and Fig. 6(b) depict the result of transmission end when x-polarized wave illuminates the metasurface. As shown in Fig. 6(a), all layers are non-working under the conditions of $E_f$ = 0.01 eV and $\sigma =10$ S/m, no focus is generated at focal plane A ($f_A$ = 4.5 mm)or focal plane B ($f_B$ = 6.5 mm). The electric field intensity at the central axis of both focal planes is essentially zero, consistent with the double-bit output state of "00". Conversely, Fig. 6(b) illustrates that setting $E_f$ = 1 eV and $\sigma =10$ S/m,the graphene layer is working and the VO$_2$ layer is non-working. Thus the x-polarized wave incident on the metasurface results in a focus at focal plane B with a focal length of 6.5 mm. The intensity of the central axis at focal plane B exhibits a notable focusing effect, while no focus is observed at focal plane A. This corresponds to the output state of CNOT LG represented by "01".
Next, Fig. 6(c) and Fig. 6(d) present the simulation result when y-polarized waves illuminate the metasurface. When the conductivity of VO$_2$ is set to $2 \times 10^5$ S/m, the relationship between the Fermi level of graphene and the high/low bits of the target bit is reversed. Specifically, when the Fermi level of graphene is 1 eV, the target bit is "0", and at this time all layers in the metasurface are working. As illustrated in Fig. 6(c), focuses can be clearly observed at both focal plane A and focal plane B under y-polarized wave incidence. It’s notable that the focus at focal plane B exhibits side lobes in the electric field intensity curve, attributed to the diffraction phenomenon of wave. Through the optimization of units and the design of numerical apertures, we have tried to reduce the SLL as much as possible. However, this does not impact the identification of high or low bits in the output, consistent with the "11" output state of CNOT LG. Importantly, according to the theory of the array antenna pattern synthesis, an additional requirement of amplitude distribution can be added to the array arrangement on the basis of phase control as shown in Eq. (4), such as Taylor distribution [55] and Schelkunoff polynomial distribution [56,57]. This method can significantly reduce the sidelobe level of the focused beam, which will greatly favors stability of the designed logic gates. In contrast, as shown in Fig. 6(d), when the Fermi level of graphene is adjusted to 0.01 eV, labeled as the target bit "1", only the VO$_2$ layer works. Consequently, a focus is generated solely at focal plane A under y-polarized wave incidence, with no focus observed at plane B. This aligns with the "10" output status of CNOT LG. Therefore, the designed THz CNOT LG metasurface for LP wave demonstrates effective functionality.
3.2 CNOT LG for CP THz wave
Arrange the units depicted in Fig. 4 into an 80x80 array according to Eq. (2) to obtain a metasurface composed of three layers of SRRs. The graphene layer forms a metalens with a focal length of 6.5 mm under normal incidence of RCP (reverse incidence of LCP), while the VO$_2$ layer corresponds to a metalens with a focal length of 4.5 mm under reverse incidence of LCP. The simulation results under four working states are shown in Fig. 7.
Fig. 7. Far-field simulation results and focus intensity distribution of xoz plane and xoy plane under different conditions: (a) $\sigma$ (VO$_2$) = 10 S/m, $E_f$ (Graphene) = 0.01 eV, RCP normal incidence. (b) $\sigma$(VO$_2$) = 10 S/m, $E_f$(Graphene) = 1 eV, RCP normal incidence. (c) $\sigma$(VO$_2$) = $2 \times 10^5$ S/m, $E_f$(Graphene) = 1 eV, LCP retro-incidence. (d) $\sigma$(VO$_2$) = $2 \times 10^5$ S/m, $E_f$(Graphene) = 0.01 eV, LCP retro-incidence. Among them, plane A is the xoy plane with z = 4.5 mm, and plane B is the xoy plane with z = 6.5 mm. When there is focus on the focal plane, the corresponding bit is "1", and when there is no focus, the corresponding bit is "0".
As shown in Fig. 7(a), setting $E_f$ = 0.01 eV and $\sigma = 10$ S/m, neither the graphene layer nor the VO$_2$ layer is working. Therefore, no focus is generated under RCP normal incidence, and the intensity of the central axis at focal plane A is nearly zero. Although certain diffraction fringes appear in the normalized intensity at the central axis of focal plane B, no focusing phenomenon is obviously observed. The "00" output state of CNOT LG can be readily identified. In contrast, Fig. 7(b) demonstrates that when only the graphene layer is working, the metasurface under RCP incidence will produce a focus within focal plane B(z = 6.5mm), corresponding to the "01" output state of CNOT LG.
As a two-parameter logic gate, the case of LCP back-incidence is recorded as a control bit "1" at the input. Due to the constraints imposed by the geometric phase conjugation, LCP should be back-incident onto the CNOT LG metasurface to induce focusing effect. When VO$_2$ is in a metallic state and the Fermi level of graphene is 1eV, all layers are working at this point. As evident from Fig. 7(c), focus is observed on both focal plane A and focal plane B, corresponding to the "11" output state of CNOT LG. Conversely, only VO$_2$ will work when the Fermi level of graphene is adjusted to 0.01 eV. As depicted in Fig. 7(d), focus is solely generated at focal plane A in this case, corresponding to the output state of "10".
Consequensly, the aforementioned metasurface for CP can effectively realize the functionalities of a THz CNOT LG gate.
3.3 CNOT performance verification based on near field imaging
Obviously, the realization of the aforementioned THz CNOT LG metasurface benefits from the actively tunable characteristics of two materials: graphene and VO$_2$. To clearly evaluate the contrast between the adjustable materials two states and the stability of the devised metasurfaces, we integrated both states of the materials into a single metasurface. Through finite element simulation, we observed the near-field imaging as shown in Fig. 8. Besides, we further analyzed the transmission spectrum and operating bandwidth of the designed units. These approach and analysis allowed for a comprehensive evaluation of the materials’ behavior and the performance of the designed metasurfaces.
Fig. 8. The near-field imaging design and results of the imaging array obtained by utilizing the "ON" or "OFF" states of the active tunable material. (a) shows the design of near field imaging metasurface for the letter "X". Utilizing the SRR in Fig. 2(a), (b) depicts the tyx versus incident wave frequency curves of the units in two states where graphene’s conductivity is "ON" or "OFF", while (c) illustrates the near-field imaging results of the letter "X". Similarly, (d) shows the design of near field imaging metasurface for the letter "Y". Employing the SRR in Fig. 2(e), (e) showcases the txy versus incident wave frequency curves of the units in two states where the conductivity of VO$_2$ is "ON" or "OFF", while (f) displays the near-field imaging results of the letter "Y". (g) shows the design of near field imaging metasurface for the letter "Y". Utilizing the SRR in Fig.4(a), (e) exhibits the tlr versus incident wave frequency curves of the units in the two states of graphene’s conductivity being "ON" or "OFF", accompanied by (i) showcasing the near-field imaging results of the letter "R". Finally, (j) shows the design of near field imaging metasurface for the letter "Y". Utilizing the unit structure in Fig.4(e), (k) demonstrates the trl versus incident wave frequency curves of units in the two states where the conductivity of VO$_2$ is "ON" or "OFF", along with (l) displaying the near-field imaging results of the letter "L".
Firstly, we resized the image with the letter "X" as depicted in Fig. 8(a) to dimensions of 50$\times$50 pixels. Next, an unit as illustrated in Fig. 2(a) at the corresponding location of each pixel was positioned. This unit comprises two identical layers of graphene and one layer of PI. As indicated in Fig. 8(a), we adjust the Fermi level of graphene on the units placed in the white part of the image to 1 eV ("ON"), while the black part corresponds to 0.01 eV ("OFF"). It is worth mentioning that the near-field imaging quality improves with a greater number of imaging arrays, corresponding to the pixel count of the image. Considering both the imaging performance and the device’s cost, we chose to design a 50$\times$50 imaging array metasurface. Upon the incidence of x-polarized terahertz waves, the units operating under the "ON" state exhibit higher transmittance to the terahertz waves, while those operating in the "OFF" state display lower transmittance. The transmission spectra of the two states at 1 THz are illustrated in Fig. 8(b). The disparity can be quantified by the modulation depth Dm, as calculated by Eq. (14). As can be inferred from Fig. 8(b), it is evident that the modulation depth of graphene-based units at 1 THz is 84.9%. This remarkable modulation depth enables the acquisition of a distinct "X" image in the near field in Fig. 8(c).
where, $T _{\rm {ON}}$ represents the cross-polarization conversion efficiency of the units in the metallic state ("ON") at 1 THz, and $T _{\rm {OFF}}$ represents the cross-polarization conversion efficiency of the meta-atoms in the insulating state ("OFF") at 1 THz.Next, as shown in Fig. 8(d), an image with a letter "Y" was sampled to 50$\times$50 pixels. The units depicted in Fig. 2(e) were positioned at both white area and black area of the image, with the conductivity of VO$_2$ set to $2 \times 10^5$ S/m ("ON") and 10 S/m ("OFF"), respectively. As Fig. 8(e), it is evident that the amplitudes of the cross-polarized components of the VO$_2$ two switching states at 1 THz are 0.4719 and 0.0006 under y-polarized wave incidence on the imaging array, respectively. Figure 8(f) illustrates the excellent near-field imaging quality in this case.
Similarly, for the near-field imaging of the letters "R" and "L", we use the units shown in Fig. 4(a) and Fig. 4(e) respectively to form a 50$\times$50 imaging array metasurface. The pixels of white area and black area are also placed with metallic and insulating units respectively, as illustrated in Fig. 8(g) and Fig. 8(j). As depicted in Fig. 8(h)-(l), the amplitude of the cross-polarized component in the "ON" and "OFF" states at 1 THz also exhibit significant differences. Clear near-field images of the letters "R" and "L" can be obtained. Notably, the unit in Fig. 8(g) is normal illuminated by the RCP wave and the unit in Fig. 8(j) is back-illuminated by the LCP wave.
Moreover, the modulation depths of tunable materials in the four aboved cases are 84.9%, 85.0%, 99.9% and 99.9%. Deeply, in the frequency range of 0.8-1.2 THz as shown in the yellow region of the transmission spectrum in Fig. 8, the modulation depths of the whole designed meta-atoms are all above 50%, which greatly reduces the frequency requirement for incident waves. Therefore, the logic gate metasurface we designed greatly reduces the requirements for the terahertz source frequency band.
In the near-field imaging work, all unit structures and the incident modes of THz waves are consistent with those of the design logic gates metasurface. These results provide strong evidence for both the performance stability of THz CNOT LG and the application of tunable materials in imaging.
4. Conclusion
In summary, we proposed a CNOT LG metasurface tailored for the THz band, addressing both cases of LP wave and CP wave. Our design employs two active tunable materials, namely graphene and VO$_2$ to establish a dual-parameter control system. By leveraging the polarization state of the incident plane wave and the distinct properties of the tunable material, we achieve a multi-input and multi-output functionality that aligns with the logical relationship of CNOT LG. Moreover, to validate the robustness of our proposed metasurface and the excellent modulation depth of the actively tunable materials, we designed an imaging array composed of tunable materials in different states, achieving ideal near-field imaging effects. This comprehensive approach not only demonstrates the performance stability of our design but also underscores the application of tunable materials in imaging. The realization of CNOT LG in the THz band holds significant practical value and opens avenues for various applications in fields such as telecommunications, sensing, and imaging.
Funding
National Natural Science Foundation of China (U22A2008); National Key Research and Development Program of China (2021YFB2800703); Laboratory Foundation, and Laoshan Laboratory Science and Technology Innovation Project (LSKJ202200801).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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