Hybrid ANN-Z Method for Modeling Carbon Nanotube-Based Reconfigurable Intelligent Surfaces for Terahertz Beam Steering

Summary

This protocol presents a hybrid artificial neural network and Z-transform method for accurate electromagnetic modeling of single-walled carbon nanotube-based optical reconfigurable intelligent surfaces operating in the terahertz band (0.5-30 THz) for 6G wireless communication applications, achieving 180× computational acceleration with reflection phase tunability exceeding 310° and beam steering range of ±45°.

Abstract

Optical reconfigurable intelligent surfaces based on single-walled carbon nanotubes offer promising solutions for terahertz beam steering and photonic wave manipulation in future 6G wireless systems. However, accurate modeling of these structures remains challenging due to quantum transport effects, kinetic inductance, and multi-resonant excitonic behavior across broad frequency ranges. This protocol describes a hybrid computational framework that integrates quantum conductivity modeling using the Kubo formalism, polynomial regression-based data smoothing, and Z-domain transfer function analysis for accurate characterization of single-walled carbon nanotube optical reconfigurable intelligent surface unit cells. The method begins with design of crossed single-walled carbon nanotube nano-strip resonators on a quartz substrate with (10,5) chirality (diameter 0.60 nm, bandgap 1.762 eV), followed by full-wave electromagnetic simulation in CST Microwave Studio across the 0.5-30 THz band. A polynomial regression model of order 8 processes the extracted S-parameters to remove numerical fluctuations and predict smoothed electromagnetic responses. A discrete transfer function H(z) with numerator order 6 and denominator order 7 is then fitted using least-squares optimization with QR decomposition, enabling pole-zero stability analysis and passivity verification. The protocol further incorporates quantum conductivity tuning via chemical potential modulation for beam steering optimization. Representative results demonstrate reflection phase tunability exceeding 310°, absorption enhancement up to 92.3%, beam steering range of ±45° with side lobe levels below -12 dB, and computational acceleration of 180× compared to conventional full-wave optimization methods. The polynomial regression achieved test root mean square error of 0.0688 with R² coefficient of 0.994, while H(z) fitting achieved root mean square error of 0.89 dB. Stability analysis confirmed all poles within unit circle. This protocol provides an efficient, reproducible pathway for designing programmable photonic metasurfaces and intelligent terahertz communication systems for 6G and beyond.

Introduction

The rapid evolution toward sixth-generation (6G) wireless systems has accelerated exploration of terahertz (THz) and optical frequency bands to achieve ultra-high data rates exceeding 1 Tbps, intelligent sensing, holographic beamforming, and adaptive wavefront engineering^1,2,3. The terahertz band (0.1-30 THz) offers abundant bandwidth but suffers from severe free-space attenuation (approximately 20 dB·km^-1 at 1 THz), atmospheric molecular absorption from water vapor at 557 GHz, 752 GHz, 988 GHz, and 1.13 THz, and extreme sensitivity to blockage from atmospheric particles and rain⁴.

Reconfigurable intelligent surfaces have emerged as an enabling technology capable of dynamically manipulating electromagnetic wavefronts through programmable reflection, absorption, phase control, and wave focusing^5,6,7. When extended to optical frequencies, these surfaces offer unprecedented control over light propagation, enabling applications such as LiDAR, free-space optical communications, holographic displays, and adaptive beam steering⁸.

Conventional reconfigurable intelligent surface structures based on metallic resonators encounter substantial limitations at terahertz and optical frequencies due to increased ohmic losses from Drude conductivity, plasmonic damping, fabrication constraints with feature sizes below 100 nm, and limited tunability of fixed dielectric properties^9,10,11. Nanomaterial-based metasurfaces, particularly those employing single-walled carbon nanotubes (SWCNTs), offer promising alternatives because of their exceptional quantum-electromagnetic properties: nanoscale dimensions with diameters of 0.7-3 nm, tunable quantum conductivity controlled by chemical potential, extraordinary carrier mobility exceeding 100,000 cm²·(V·s^-1), high thermal conductivity of approximately 3000 W·(m·K^-1), and strong excitonic resonances in visible and near-infrared regimes^12,13,14,15. SWCNTs exhibit strong electromagnetic interactions in terahertz and photonic regimes through exciton-photon coupling, enabling compact optical resonators with programmable electromagnetic responses¹⁶. These properties make SWCNTs attractive candidates for optical reconfigurable intelligent surfaces, where compact size, tunable response, and efficient wave manipulation are important design requirements.

Despite these advantages, accurate modeling of SWCNT-based optical reconfigurable intelligent surfaces remains challenging due to quantum transport effects requiring non-equilibrium Green's function approaches, kinetic inductance dominating at frequencies above 100 GHz, multi-resonant excitonic behavior with binding energies up to 0.4 eV, nonlinear dispersion phenomena arising from electron-phonon coupling, chirality-dependent optical properties requiring specification of chiral indices (n, m), and temperature-dependent conductivity requiring self-consistent thermal modeling^17,18,19,20. Traditional equivalent circuit approaches and conventional electromagnetic fitting methods such as rational approximation and vector fitting often fail to accurately represent these complex interactions over broad frequency ranges covering 0.5-30 THz, a 60× bandwidth^21,22,23. These limitations can reduce modeling accuracy and increase computational complexity during the design of terahertz reconfigurable intelligent surfaces, highlighting the need for efficient modeling approaches that preserve physical interpretability while accurately capturing broadband electromagnetic behavior.

This protocol introduces a hybrid modeling framework that uniquely combines quantum conductivity analysis using the Kubo formalism with chirality-dependent bandgap calculation, polynomial regression for noise reduction and response smoothing, Z-transform transfer function analysis for pole-zero stability and physical interpretability, and full-wave electromagnetic simulation of SWCNT-based unit cells. The workflow provides a step-by-step procedure for constructing, analyzing, and optimizing SWCNT-based optical reconfigurable intelligent surfaces across the terahertz frequency range. The method is designed for researchers and engineers working on advanced electromagnetic surfaces, nanophotonics, and next-generation wireless communication systems. The protocol enables users to generate stable and physically interpretable models of SWCNT-based optical reconfigurable intelligent surfaces for beam-steering and electromagnetic wave-control applications. The protocol assumes familiarity with electromagnetic simulation concepts but provides detailed steps for replication by researchers new to the field.

Protocol

1. SWCNT Optical RIS Unit Cell Design

1. Selection of SWCNT Chirality
   1. Select SWCNT chirality (10,5) based on quantum conductivity analysis.
   2. Calculate the nanotube diameter using below mentioned formula
      
      where a_cc=0.142 nm is the carbon-carbon bond length. The (10,5) chirality yields diameter of 0.60 nm and bandgap of 1.762 eV, optimal for terahertz operation.
   3. Calculate the chiral angle using
2. Unit Cell Geometry Definition
   1. Design crossed SWCNT nano-strip resonators on a quartz substrate with the following parameters: substrate permittivity of 3.8, substrate thickness of 500 nm, unit cell dimensions of 1.5 × 1.5 µm², SWCNT strip width of 50 nm, SWCNT strip length of 700 nm, and inter-strip spacing of 120 nm. These dimensions ensure sub-wavelength operation across the 0.5-30 THz band.
3. CST Microwave Studio Simulation Setup
   1. Launch CST Microwave Studio and create a new project using the Microwave and RF frequency domain solver.
   2. Construct unit cell geometry using defined parameters. Apply unit cell boundary conditions in the x and y directions with periodic boundaries. Define Floquet ports along the z-direction for plane wave excitation.
   3. Set the frequency sweep range from 0.5 THz to 30 THz with a step size of 0.05 THz, generating 590 frequency points. Configure the time-domain solver with adaptive mesh refinement and set convergence target to -40 dB.
   4. Run the full-wave electromagnetic simulation. Export the complex S₁₁ reflection coefficient, S₂₁ transmission coefficient, reflection phase, and absorption spectra as CSV files.

2. Quantum Conductivity Modeling Using Kubo Formalism

1. Physical Constants Initialization
   1. Initialize physical constants: elementary charge e=1.602×10⁻¹⁹C, reduced Planck constant ħ=1.0546×10⁻³⁴J·s, Boltzmann constant k_B=1.3806×10⁻²³J·K^-1, temperature T=300K, Fermi velocity v_F=8×10⁵m·s^-1, and free-space impedance η₀=377Ω.
2. Intra-band Conductivity Calculation
   1. Calculate the intra-band conductivity for each frequency point using:
      
      where τ=0.5ps is carrier relaxation time and E_F is Fermi energy (varied from 0.1e.V to 0.4eV).
3. Inter-band Conductivity Calculation
   1. Calculate the inter-band conductivity using:
      
   2. Compute total surface conductivity as σ_total(ω) = σ_intra(ω) + σ_inter(ω).
4. Optical Response Calculation
   1. Calculate substrate phase delay for quartz (ε_r=3.8, thickness=500nm) using φ_sub=2πf(n_sub)d_sub/c, where n_sub=.
   2. Compute reflection coefficient from conductivity using
      
   3. Calculate reflectance as R(ω)=|S₁₁(ω)|². Calculate absorption as A(ω)=1-R(ω)-T(ω).

3. CST Data Generation and Preprocessing

1. Emulated CST Data Generation
   1. Generate emulated CST full-wave simulation results by adding realistic resonances to the quantum S₁₁.
   2. Define five resonances: E₁₁ exciton at 2.8 THz (magnitude -15.2 dB, phase 45°), E₂₂ exciton at 5.6 THz (-22.8 dB, -120°), plasmonic resonance at 12.4 THz (-8.5 dB, 60°), cavity mode at 18.9 THz (-12.1 dB, -30°), and phonon-assisted resonance at 24.7 THz (-6.8 dB, 15°).
   3. Add numerical noise with noise level 0.03 (SNR=30.5dB) to simulate CST numerical fluctuations.
2. Data Preprocessing
   1. Normalize frequency axis to [0,1] using
      
   2. Extract real and imaginary parts of S₁₁ and magnitude of S₂₁.
   3. Split dataset into training (80%), validation (10%), and testing (10%) using random permutation. Normalize inputs to zero mean and unit variance.

4. Polynomial Regression for Data Smoothing

1. Fit an 8^th order polynomial to the real part of S₁₁ using polyfit: P_real = polyfit(X_train, Y_train(:,1), 8). Fit an 8^th order polynomial to the imaginary part: P_imag = polyfit(X_train, Y_train(:,2), 8). Fit an 8^th order polynomial to S₂₁ magnitude: P_mag = polyfit(X_train, Y_train(:,3), 8). Generate smoothed predictions using polyval on full frequency range. Compute smoothed S₁₁ as S₁₁_ann = S11_real_ann + i × S11_imag_ann.

5. Z-Domain Transfer Function Fitting

1. Set sampling frequency F_s=60THz (2×maximum frequency per Nyquist criterion). Map frequencies to Z-domain using z=e^{i2πf/Fs}.
2. Define discrete transfer function in the Z-domain is expressed as:
   
   where numerator order n=6 and denominator order m=7.
3. For Least-Squares Optimization, set up linear equations A × x= B, where A contains numerator and denominator terms and B contains ANN-predicted S₁₁. Solve using QR decomposition for numerical stability:
   
4. Pole-Zero and Stability Analysis
   1. Extract poles by solving denominator polynomial. Ensure stability by projecting poles with |pole| ≥ 1 inside unit circle using poles(p) = poles(p) / (|poles(p)| + 0.1).
   2. Reconstruct denominator from stabilized poles. Extract zeros by solving polynomial numerator. Evaluate H(z) on frequency grid and calculate root mean square error.

6. Beam Steering Optimization

1. The normalized array factor for a linear phased array is given by:
   ​
   where I_n = 1 for uniform excitation, k = 2π/λ, d = λ/2 spacing, β_n is progressive phase shift. For target steering angles of ±45°, ±30°, and 0°, calculate phase shift between adjacent elements as .EQUATION
   1. Compute far-field patterns and evaluate side lobe levels and half-power beamwidth.

Results

Selection of SWCNT Chirality
The protocol described was implemented for a (10,5) SWCNT optical RIS unit cell operating across the 0.5-30 THz band. Representative results demonstrate the effectiveness of the hybrid polynomial-Z modeling approach for accurate electromagnetic characterization and beam steering optimization.

Quantum Conductivity Analysis
Kubo formalism revealed that the (10,5) SWCNT exhibits complex surface conductivity dominated by imagin...

Discussion

Critical steps in the protocol require careful attention to ensure successful implementation. First, accurate selection of SWCNT chiral indices is essential because the bandgap and optical response are highly chirality-dependent. The (10,5) chirality specified in this protocol provides optimal bandgap of 1.762 eV for terahertz operation, but users targeting different frequency bands should calculate the corresponding chirality using the bandgap formula E_g = 2ħv_F/d = (2 × 1.0546×10⁻³⁴

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
CST Microwave Studio	Dassault Systèmes	N/A	Version 2024, Frequency domain solver
MATLAB	MathWorks	N/A	Version R2014a or later
Quartz substrate	University Wafer	4526	500 nm thickness, ε_r = 3.8
SWCNT (10,5) chirality	NanoIntegris	SWCNT-105	0.60 nm diameter, >90% semiconducting
Personal Computer	N/A	N/A	32 GB RAM, 8 CPU cores minimum