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Analog Computing Using Graphene-based Metalines

Sajjad AbdollahRamezani, Kamalodin Arik, Amin Khavasi, Zahra Kavehvash

arXiv:1509.01226v1cs.ETcond-mat.mes-hallphysics.optics

TL;DR

The paper presents a planar graphene-based meta-transmit-array for wave-based analog computing. Simulated differentiation and integration outputs closely track desired results, and the first-derivative response has smaller amplitude and phase deviations than the cited prior structure.

Abstract

from arXiv · show

We introduce the new concept of "metalines" for manipulating the amplitude and phase profile of an incident wave locally and independently. Thanks to the highly confined graphene plasmons, a transmit-array of graphene-based metalines is used to realize analog computing on an ultra-compact, integrable and planar platform. By employing the general concepts of spatial Fourier transformation, a well-designed structure of such meta-transmit-array combined with graded index lenses can perform two mathematical operations; i.e. differentiation and integration, with high efficiency. The presented configuration is about 60 times shorter than the recent structure proposed by Silva et al.(Science, 2014, 343, 160-163); moreover, our simulated output responses are in more agreement with the desired analytic results. These findings may lead to remarkable achievements in light-based plasmonic signal processors at nanoscale instead of their bulky conventional dielectric lens-based counterparts.

THEORETICAL FRAMEWORK

The framework treats the system as linear and transversely invariant, relating input and output through convolution. In one dimension, Fourier-domain transfer functions specify operations; graded-index lenses provide the Fourier-transform step, while a double transform yields a mirrored output.

THEORETICAL FRAMEWORK

  • The system assumes linear transverse invariance, with an arbitrary input function and its output related through linear convolution.The impulse response is the desired two-dimensional response.Source: THEORETICAL FRAMEWORK · p. 3
  • Fourier transforming the input-output relation expresses it using the transforms of the input, output, and impulse response.The framework identifies these as F(kx,ky), G(kx,ky), and H(kx,ky), respectively.Source: THEORETICAL FRAMEWORK · p. 3
  • For the graphene-based system, the formulation reduces to one transverse dimension, where H(kx) is the transfer function between incident and transmitted field profiles.The fields are represented by f(x) and g(x).Source: THEORETICAL FRAMEWORK · p. 3
  • A lens performs the Fourier transform, and applying a second transform produces an output proportional to the mirror image of the desired function.The second transform is used because realizing an inverse Fourier transform with real materials is not possible.Source: THEORETICAL FRAMEWORK · pp. 3–4
  • Graphene-based graded-index lenses approximate dielectric lenses by patterning conductivity so graphene-plasmon mode index follows a quadratic refractive-index profile.The gradual optical-property change of GRIN lenses can significantly reduce scattering and increase efficiency.Source: THEORETICAL FRAMEWORK · p. 4

GRAPHENE-BASED META-TRANSMITT-ARRAY

A symmetric three-metaline graphene meta-transmit-array is designed to control transmitted-wave amplitude and phase locally. Its unit cells can achieve any transmission phase and amplitude profile through chemical-potential tuning.

GRAPHENE-BASED META-TRANSMITT-ARRAY

  • A symmetric stack of three metalines, with quarter-guided-wavelength spacing, controls transmission amplitude and phase independently.The two outer stacks are identical; their spacing from the inner stack minimizes transmission-amplitude ripples.Source: GRAPHENE-BASED META-TRANSMITT-ARRAY · p. 4
  • The array manipulates graphene-plasmon propagation along and across it by varying graphene surface conductivity with an uneven ground plane.This enables local control of transmission phase in addition to amplitude.Source: GRAPHENE-BASED META-TRANSMITT-ARRAY · p. 4
  • The building-block scattering parameters are calculated from a transfer matrix formed by multiplying interface-matching and segment-propagation matrices.The interface treatment uses reciprocity; the propagation matrix relates forward and backward fields along a segment.Source: GRAPHENE-BASED META-TRANSMITT-ARRAY · p. 5
  • Local tuning of inner and outer metaline chemical potentials can achieve any transmission phase and amplitude profile.The amplitude and phase of S21 are plotted against those chemical potentials.Source: GRAPHENE-BASED META-TRANSMITT-ARRAY · p. 6

REPRESENTATIVE RESULTS

The designed transfer functions implement first- and second-order differentiation and second-order integration. Simulations show close agreement with desired or analytical outputs, with a reported amplitude and phase deviation advantage over Silva et al. for the first-order differentiator.

First-order differentiation

  • The first-order differentiator uses a transfer function proportional to (ix/(W/2))^n, normalized so maximum transmittance is unity.The lateral chemical-potential profile implements the transfer function's transverse amplitude and phase distributions.Source: REPRESENTATIVE RESULTS · p. 6
  • The designed meta-transmit-array output profile agrees excellently with the desired transfer function, with amplitude and phase standard deviations of 0.04 and 6°, below Fig. S7b of [1] (about 0.19 and 15°).Accordingly, a more accurate output response can be anticipated for the whole structures.Source: REPRESENTATIVE RESULTS · p. 7

Second-order differentiation

  • The second-order differentiator uses H(x) proportional to −(x/(W/2))^2, with quadratic amplitude and constant phase.The simulated results are illustrated in Figure 7.Source: REPRESENTATIVE RESULTS · p. 7

Second-order integration

  • For second-order integration, the ideal inverse-square transfer function diverges near x = 0, so the design approximates it with a central unity-amplitude region.Within |x| < h the amplitude is unity; outside, it follows the desired profile, with h = W/12. The simulated output agrees excellently with the analytical solution.Source: REPRESENTATIVE RESULTS · p. 7

Operator demonstrations

  • With a Sinc-function input, simulations of the first-order derivative, second-order derivative, and second-order integrator outputs are closely proportional to their desired results.The reported configurations are GRIN lens/meta-transmit-array/GRIN lens systems.Source: REPRESENTATIVE RESULTS · p. 7

CONCLUSIONS AND OUTLOOK

The paper proposes a planar, three-metaline graphene meta-transmit-array for wave-based analog computing, using surface-conductivity tailoring to control transmission amplitude and phase. The differentiator and integrator show high efficiency and more accurate responses than the bulky structure in [1], while fabrication imperfections and tolerance may distort results and degrade efficiency.

Conclusions and outlook

  • The proposed planar array uses three symmetric stacked graphene metalines to perform wave-based analog computing through tailored surface conductivity and spatial-Fourier-domain transfer functions.The designed differentiator and integrator are reported to have high efficiency and more accurate responses than the bulky structure in [1].Source: CONCLUSIONS AND OUTLOOK · p. 8
  • The miniature structure's performance is sensitive to future fabrication imperfections and tolerance, which the authors say will distort anticipated results and degrade efficiency.The authors suggest the approach may support more complex nanoscale signal processors.Source: CONCLUSIONS AND OUTLOOK · p. 8

MATERIALS AND METHODS

The study uses full-wave electromagnetic simulations of graphene modeled through its surface conductivity and impedance. Simulations use specified material conditions and dimensions, and proceed block by block because of the small mesh size.

Simulation setup

  • Full-wave simulations use Ansoft HFSS and model graphene as a surface impedance Zs = 1/σs.The conductivity is obtained from the Kubo formula under the random phase approximation.Source: MATERIALS AND METHODS · pp. 8–9

Material assumptions

  • Simulations assume a free-standing graphene layer, with temperature T = 300 K and relaxation time τ = 1 ps.The authors describe these values as relevant to recent experimental achievements.Source: MATERIALS AND METHODS · pp. 8–9

Simulation procedure

  • To reduce computational cost from the very small mesh size, the GRIN lens, meta-transmit-array, and second GRIN lens are simulated block by block using each block's output as the next input.A segmented current sheet supplies the desired incident field distribution.Source: MATERIALS AND METHODS · p. 9

Geometry and wavelength

  • The simulated structure uses W = 684 nm, Lg = 1028 nm, D = 100 nm, Λ = 18 nm, d = 5 nm, and λ = 6 µm.The authors state that other wavelengths are possible when constituent parameters and dimensions are designed appropriately and graphene supports plasmons at that wavelength.Source: MATERIALS AND METHODS · pp. 9–10

Transverse Chemical Potential Profile

The figures show the first-order differentiator’s transverse chemical-potential and conductivity profiles, alongside simulated field and output comparisons for differentiation and integration.

Transverse Chemical Potential Profile

  • Figure 5 shows the transverse chemical-potential distribution and corresponding complex surface conductivity for the first-order differentiator.Source: Figure 5 · p. 16
  • Figures 6 and 9 show first-order differentiation: transmitted amplitude and phase are compared with the desired transfer function, and output-field components with analytical results.The input for Figure 9 is f(x) = sinc(16πx/W).Source: Figure 6 · p. 17, Figure 9 · p. 20
  • Figures 7 and 10 show second-order differentiation, comparing transmitted amplitude and phase with the desired transfer function and output-field components with analytical results.The input for Figure 10 is f(x) = sinc(16πx/W).Source: Figure 7 · p. 18, Figure 10 · p. 21
  • Figures 8 and 11 show second-order integration, comparing transmitted amplitude and phase with the desired transfer function and output-field components with analytical results.The input for Figure 11 is f(x) = sinc(16πx/W).Source: Figure 8 · p. 19, Figure 11 · p. 22
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