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Abstract
Under the trends of multifunctionality, tunability, and compactness in modern wave-based signal processors, in this paper, we propose a polarization-multiplexed graphene-based metasurface to realize distinct mathematical operators on the parallel time-domain channels enabled by vertical and horizontal polarizations. The designed metasurface is composed of two perpendicularly-oriented graphene strips for each of which the chemical potential can be dynamically tuned through a DC biasing circuit. The programmable metasurface exhibits two orthogonal channels through which the time-domain input signals are elaborately processed by separate mathematical functions. Several illustrative examples are presented demonstrating that the proposed device can operate on different time-domain analog computing modes such as fractional-order differentiator and phaser at the same time. The strategy introduced in this paper will enable real-time parallel temporal analog computing and has potentially essential applications in terahertz spectroscopy architectures, communication systems, and computing technologies.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the last few decades, digital signal processors (DSPs) have received significant attention in performing computational tasks, because of their flexibility, stability, and reliability [1], [2]. However, they have some drawbacks, such as restricted processing speed, excessive power consumption, and incompatibility with high-frequency operations due to limitations associated with the digital-to-analog converters [3]. Recently, the urgent need to high-speed, low-loss, and compact architectures has triggered some new research efforts to develop analog computing platforms [4–6]. The growing development of artificial materials brought the wave-based analog signal processing (ASP) back to the competition [7], which can realize real-time and parallel computations with low energy consumption [3]. The implementation of different mathematical operations like spatial differentiation, integration, or convolution has been recently demonstrated in the either spatial [7–13] or temporal [14–21] analog computing domains.
Within the spatial processing frameworks, categorized in Metasurface [22–25] and Green’s function [26–35] approaches, the input signals are depicted by the transversal shape of the input fields in the spatial domain, where the angular response of the computing device builds the transfer function of interest [7]. In the temporal analog computing, however, the input signals can be thought of as the time content of the input fields, which can be elaborately manipulated by the spectral response of the processing structure [33]. Although many efforts have been paid to enrich spatial signal processing strategies [3], [8] the great achievements of artificial structures have not been fully exploited in wave-based temporal computing, and there are still many aspects needing improvement in this area. For instance, Pandian and Seraji have set the basis of modern optical temporal analog signal processing, showing that the transient optical response of a fiber ring resonator can be used for various operations such as pulse delay, differentiation, and integration [34]. Over recent years, a few designs [33], [35–41] have been proposed for performing analog signal processing in time such as differentiation and integration, and differential equation solving [42]. For example, a fractional-order photonic temporal differentiator was proposed in [38] based on the silicon hybrid plasmonic add-drop microring resonators. The group of authors in [39] investigated a miniaturized broadband silicon hybrid plasmonic integrator in the temporal domain with the functionality for solving ordinary differential equations. Lie et. al. designed a phase-shift fiber Bragg grating to perform the time differential and integral of the complex envelope of one input optical signal at the same time [33]. Despite all these efforts, there are barriers to benefit parallel signal through using the previous temporal computing proposals. Several studies have already investigated the potential application of programmable metasurfaces to realize time-domain signal processing [19,20]. The authors in [19] experimentally demonstrated a heuristic approach for implementation of over-the-air time-domain signal processing through a programmable metasurface. Hougne et.al. [20] have shown that a reconfigurable metasurface suffices to perform temporal analog computation with Wi-Fi waves reverberating in a room. Although frequency multiplexing [19,13] and real-time programming [20] were elaborately exploited to aim this goal, polarization-multiplexed programmable computing channels have not been investigated within the temporal domain, yet. Moreover, the mentioned studies have been conducted at microwave frequencies. Due to some reasons like material losses, designing a similar programmable processor at higher frequencies has usually more challenges in comparison to their counterparts in the microwave regime.
In this paper, an anisotropic programmable metasurface is proposed to provide two TE- and TM-polarized parallel temporal processing channels at THz frequencies. The designed metasurface is made of two perpendicularly-oriented graphene ribbons for each of which the chemical potential can be dynamically altered via a specifically-designed biasing circuit. We demonstrate that different computing tasks such as fractional-order differentiation and group-delay editing can be simultaneously accomplished through the TE- and TM-polarized channels of the designed metasurface. This work opens the avenues for designing multifunctional temporal analog signal processors with parallel computing ability to handle separate parts of an overall task and accelerate the processing. Such a programmable wave-based metasurface may be a key component in dual-polarized terahertz spectroscopy architectures where the post-processing computing tasks can be directly performed by the proposed processor.
2 Programmable meta-atom design
As the main goal of this paper, the programmable metasurface should realize two independent processing channels for each of which the spectral transfer function is dynamically controlled. As shown in Fig. 1(a), the time-domain computing TE and TM channels are occupied by the TE- and TM-polarized components of the input field, respectively. In order to have different spectral transfer functions in these channels, an anisotropic graphene-based sub-wavelength meta-atom is elaborately designed (see Fig. 1(b)) comprising of two perpendicularly-oriented graphene strips having controllable chemical potential values. Each graphene strip is etched on a Quartz (εr = 3.75, tanδ=0.0004) substrate with the thickness of 1 µm and the overall structure is also placed over another Quartz layer with the thickness of 1 µm. To avoid energy transmission, a gold layer ($\sigma = 4.56 \times {10^7}$S/m) of thickness 5 nm terminates the meta-atom. The graphene strip, a polysilicon substrate with the thickness of 100 nm, an Alumina layer with the thickness of 65 nm, and top/bottom contacts made of Cr/Au make a gated structure in each meta-atom [43]. The thickness of auxiliary layers is much smaller than the operating wavelength so that their impact on the phase and amplitude of reflection spectrum can be safely ignored. The programmability of the designed meta-atoms is originated by the tunable performance of the graphene layers, where the chemical potentials of graphene strips in each meta-atom can be separately adjusted through practical technologies such as electrical gating. The external biasing voltages can be differently applied between the polysilicon layer and conductive contacts for each graphene strip (see Supplement 1, Appendix A). Indeed, the graphene layers are geometrically identical while characterized with different Fermi energy levels.
Fig. 1. The schematic illustration of (a) the metasurface-based temporal processor and (b) the designed programmable meta-atoms (c) the proposed test setup for our polarization-dependent device.
The overall metasurface is occupied by 16 × 16 meta-atoms whose electromagnetic response upon illuminating by TE- and TM-polarized radiations are dynamically manipulated without any information cross-talk. Two different DC signals are utilized, one of which, bias all horizontally-aligned graphene layers while the other is connected to all vertically oriented graphene stripes.
Interestingly, the electromagnetic response of the metasurface to a y-polarized illumination is sensitive to the plasmonic resonances in the y-direction that are only affected by µc, y (the chemical potential of y-directed strip). While, x-polarized incident waves sense the plasmonic resonances of the metasurface along the x- direction which are only affected by µc, x (the chemical potential of x-directed strip). It should be noted that although the graphene strips in each meta-atom are differently fed, the co-oriented graphene layers share identical biasing voltages. Choke filters can be also utilized to eliminate the flow of the THz currents to the DC biasing circuit [44]. Therefore, the designed metasurface is a worthy candidate to provide two real-time temporal processing channels (TE/TM channels) with distinct user-defined transfer functions. The periodicity of the structure is 10 µm along x- direction and 9 µm along y- direction and the other geometrical parameters have been optimized so that the metasurface exposes our desired transfer function in each processing channel.
Graphene can be considered as a flat mono-atomic layer of carbon atoms bonded in a hexagonal structure [45], [46] with the complex surface conductivity expressed by the Kubo formula [46], [47]:
Here, ${\hbar}$ and ${k_B}$ parameters are stand for the reduced Planck’s and Boltzman’s constants, respectively, ${\mu _c}$ denotes the chemical potential, e refers to the electron charge, $\tau $ is the relaxation time, T indicates the environmental temperature, and $\omega $ shows the angular frequency. In our numerical simulations, the graphene layers are modelled as complex surface impedances with the chemical potential range between 0.2 eV to 1.2 eV, the relaxation time is 0.1 ps, and T = 300 K.
3. Illustrative examples
This paper exploits parallel temporal analog computing in which several time-domain operations such as differentiation and phaser can be realized over two different channels at the same time. The transfer functions associated with these channels, $\widetilde H(\omega - {\omega _0})$, are the spectral information of the Rxx and Ryy coefficients of the anisotropic graphene-based metasurface in which ${\omega _0}$ denotes the central frequency. The incident and reflected electric fields, expressed in the time domain, are denoted as ${E_{in}}(t) = \int {{\widetilde E_{in}}(\omega - {\omega _0})\exp (j\omega t)d\omega } $ and $E_{out}(t) = \int {{\widetilde E_{out}}(\omega - {\omega _0})\exp (j\omega t)d\omega } $, respectively. The output signal of the system can be expressed as
in which, the tilde notation indicates the Fourier transform of signals and F-1 stands for the inverse Fourier transform. Therefore, the temporal transfer function can be written byIn this section, we intend to design a single metasurface implementing two different temporal processing tasks in the temporal domain: (i) first-order temporal differentiator over the TM-polarized channel and (ii) pulse-position modulation over the TE-polarized channel. For a better understanding of the concept, a schematic configuration for the test setup of the proposed analog computing system is displayed in Fig. 1(c). A circularly-polarized beam impinges on the programmable metasurface processor; then, the reflected beam excites the polarizer so as to separate the TE- and TM-polarized processed signals. The processed beams are received by two proper cameras.
3.1 Pulse-envelope differentiation/pulse spreading
The transfer function of an ideal first-order differentiation can be defined as
Here, $\alpha $ is a constant value denoting the gain of differentiator. Thus, the reflection coefficient spectrum of the metasurface should experience a π sharp phase jump while maintaining a linear amplitude in the vicinity of ${\omega _0}$ with ${\widetilde H_{diff}}({\omega _0})=0$.
Let us consider an optical pulse with the envelope of A0(t) and the central frequency ${\omega _0}$ impinges normally on the programmable metasurface, ${E_{in}}(t) = {A_0}(t)\exp ( - j{\omega _0}t)$. The first-order differentiation operates for the pulse envelope as
The schematic sketch of the temporal differentiation functionality through the proposed metasurface-based processor is shown in Fig. 2.
On the other hand, a spatial phaser [48] is a linear time-invariant system with ${\widetilde H_{phaser}}(\omega - {\omega _0}) = b\exp (j\varphi (\omega - {\omega _0}))$, where b denotes a positive constant and
Consequently, the group delay refers to the time delay of the amplitude envelope when the input field is reflected by the designed metasurface. The pulse-spreading function endows us with the ability to steer the amplitude envelopes of quasi-sinusoid waves as desired. Let us consider an input signal that travels through a metasurface phaser with a linear group-delay response
Hereafter, we intend to design a metasurface processor with TE and TM channels dedicated to realizing pulse spreading and first-order differentiation, respectively. The geometrical parameters of the structure are optimized so that the complex ${\widetilde R_{_{TE}}}$ and ${\widetilde R_{_{TM}}}$ coefficients mimic the spectral information of ${\widetilde H_{phaser}}$ and ${\widetilde H_{diff}}$ around f0 = 5.6 THz, respectively. The numerical simulations have been executed by using the electromagnetic commercial software, CST Microwave Studio, where the periodic boundary conditions have been applied to the lateral directions and Floquet boundaries have been assigned along z-direction. Figure 4 depicts the synthesized temporal transfer functions of the optimized programmable metasurface, i.e., ${\widetilde R_{_{TM}}}$ and ${\widetilde R_{_{TE}}}$, for normal incidence. The best structural parameters are given in the caption of the same figure. As a reference, the ideal results are also plotted in the same figure.
Fig. 4. The amplitude and phase spectra of the reflection coefficient for TM polarization (TM channel), realizing the transfer function of a first-order temporal differentiator. The optimized parameters are; w = 4 µm, l = 21 µm, a = 4.5 µm, b = 8 µm, t = 1 µm, Px = 10 µm, and Py = 9 µm.
As noticed from Fig. 4, $|{{{\widetilde R}_{_{TM}}}} |$ curve experiences a sharp dip at f0 = 5.6 THz and a linear trend with the gain parameter a = 0.5 THz-1 while $\measuredangle {\widetilde R_{_{TM}}}$ undergoes $\pi $ phase jump in the vicinity of the carrier frequency. The minimum amplitude of the transfer function is about $- $46 dB (See Supplement 1, Fig. S3). The asymmetry of the reflection amplitude may be attributed to the frequency-dependent graphene losses originating from the propagation of surface plasmon polaritons [49]. However, the required asymmetry and the linear trend of the reflection amplitude are both satisfied over the specified frequency bandwidth around the center frequency. Considering an incident electromagnetic pulse (see Fig. 5(a)) with the envelop satisfying the Gaussian function $\exp ( - {t^2}/{\delta ^2})\exp ( - j{\omega _0}t)$ in which $\delta $ refers to the pulse width (the temporal FWHM is $({(2\ln (2))^{1/2}}\delta )$), by applying ${\widetilde R_{_{TM}}}$ transfer function in the Fourier domain, the time-domain reflected pulse is calculated and displayed Fig. 5(b) (for FWHM = 15 µs). As observed, a slight difference between the envelopes of the output pulse of the actual metasurface and that of the ideal derivative case verifies the performance of the horizontal processing channel. The frequency bandwidth of TM channel for performing the first-order differentiation is about 2THz. Figures 6(a), (b) illustrates the amplitude/phase and group delay of ${\widetilde R_{_{TE}}}$ coefficient for the designed metasurface compared with the benchmark solutions, respectively. As seen, the synthesized group delay approximately rises linearly from 4.7 to 5.753 THz while shows a negative constant slope between 5.753 to 6.7 THz. Due to the loss of the graphene material, a small dip is obtained in the reflection amplitude near f0 = 5.6 THz. The frequency bandwidth of TE channel for realizing the triangular-shape group delay is about 2 THz.
Fig. 5. (a) The envelope of the input Gaussian pulse for TM polarization (TM channel). (b) The simulated and exact output signal of the designed temporal differentiator.
Fig. 6. (a) The phase and amplitude spectra of the designed phaser for TE-polarized incident waves (TE channel). (b) The group delay of the metasurface processor. The optimized parameters are; w = 4 µm, l = 21 µm, a = 4.5 µm, b = 8 µm, t = 1 µm, Px = 10 µm, Py = 9 µm, and $\mathrm{\Delta \tau =\ \beta \Delta \omega =\ 0}\textrm{.87ps}$.
3.2 Fractional-order temporal differentiation
The fractional derivative is a concept in mathematics to describe compact dynamic systems behavior in micro and nano scales with important applications biomedicine, image/signal processing, and signature verification [50]. An n-order temporal differentiation operation results in n-order time derivative of the complex envelope of the input field, i.e., ${d^n}A(t)/d{t^n}$. Here, n is a non-integer positive number. Thus, the required transfer function can be described as [51]
As can be seen, the processing channel of the programmable metasurface should be engineered to show an amplitude response of ${|{(\omega - {\omega_0})} |^n}$ and phase response of ± nπ/2, near the carrier frequency ${\omega _0}$. We seek for the best group of the structural parameters whereby both ${\widetilde R_{_{TM}}}$ and ${\widetilde R_{_{TE}}}$ coefficients of the graphene-based metasurface emulate the spectral trend of Eq. (11) but with a different order, n. For different values of the chemical potentials, the amplitude and phase spectra of ${\widetilde R_{_{TM}}}$ and ${\widetilde R_{_{TE}}}$ coefficients are plotted in Figs. 7(a), (b), respectively. For the sake of simplicity, the chemical potential of both graphene strips in the meta-atoms are selected identically, resulting in ${\widetilde R_{_{TM}}} = {\widetilde R_{_{TE}}}$. These values can be selected differently when needed to control both TM and TE channels separately. The optimized parameters are given in the caption of the same figure. As displayed, by altering the chemical potential, the phase jump is changed from 110 to 157 degree, leading to the fractional differentiation order of n = 0.61, 0.67 and 0.87.
Fig. 7. (a) The amplitude and (b) phase spectra of the designed fractional-order differentiator for TM polarization (TM channel). The optimized parameters are; w = 4 µm, l = 21 µm, a = 4.5 µm, b = 8 µm, t = 1 µm, Px = 10 µm, and Py = 9 µm.
The results for different chemical potential values are presented in Supplement 1, Fig. S4. Meanwhile, the designed metasurface provides ${|{\omega - {\omega_0}} |^n}$ (where n ϵ [0.61,0.87]) amplitude dependence around the carrier frequency of f = 5.6 THz. Therefore, the reflection response of the proposed processor is in a good agreement with that of the fractional-order differentiator within the frequency bandwidth of 2 THz around the incident frequency. Figures 8(a)–(d) illustrate an input Gaussian pulse with FMWH =1 µs, centered at f = 5.6 THz, and the differentiated output pulse of the designed programmable metasurface with different values of n. For the sake of comparison, the same order differentiation results of the input pulse are also calculated analytically and presented in the same figure.
Fig. 8. (a) The envelope of the input signal and the corresponding output signals of the designed fractional-order differentiator for TM polarization (TM channel) with (b) n = 0.61, (c) n = 0.67, and (d) n = 0.87.
Two important features of the reflected signals are the ratio of the first and the second peaks and the zero point of the differentiated pulse, match with those of the benchmark solutions, validating the performance of the fractional-order temporal differentiator. It should be mentioned that each of ${\widetilde R_{_{TM}}}$ and ${\widetilde R_{_{TE}}}$ coefficients can be programmed to show such spectral responses depending on whether the chemical potential of the horizontally- or vertically-oriented graphene layers is equal to the optimized value, respectively. Note that all temporal processing functionalities presented in this paper can be dynamically switched to each other through changing only the biasing voltages of the vertically- and horizontally-oriented graphene strips [52–55].
At the end, we should note that the synthesis of arbitrary temporal waveforms as the input signal of the proposed programmable device can be realized in different ways such as diffractive surfaces [56] or optical mapping into the Fourier plane [57].
4. Conclusion
This paper proposed a dual-channel programmable metasurface processor to realize parallel signal processing in the temporal domain. The designed metasurface is based on graphene-based meta-atoms with two vertical layers controlled by distinct biasing voltages. The designed programmable metasurface exhibits two differently-polarized processing channels whose transfer functions can be dynamically adjusted through changing the biasing voltages of each graphene layer. Illustrative examples were presented to show that the designed temporal processor can accomplish diverse analog computing tasks (first-order differentiation, phaser, fractional-order differentiation) at two differently-polarized channels at the same time. To the best of authors’ knowledge, this paper is the first proposal of programmable time-domain processors with parallel computing ability based on metasurfaces. The numerical results in all of the cases mentioned above are in a good agreement with the ideal cases. The unique properties of the proposed programmable processor suggest its application in the ultrafast all-optical analog computing systems, dual-polarized terahertz spectroscopy architectures (See Supplement 1, Appendix D), and time-domain imaging platforms.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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