A robust and efficient method to calculate electromagnetic modes on a cylindrical step-index nanofibre

1. Problem Statement

The accurate calculation of guided electromagnetic modes in optical nanofibres is essential for applications in nanophotonics, including quantum interfaces, chiral quantum optics, and vectorial light sensing.

• The Conventional Approach: Standard textbook methods involve solving Maxwell's equations in the core and cladding, enforcing continuity of tangential fields ($E_z, H_z, E_\phi, H_\phi$) at the boundary. This results in a homogeneous linear system of four equations with four unknown constants ($A, B, C, D$), represented as a $4 \times 4$ matrix eigenvalue problem.
• The Flaw: While the dispersion relation (finding the propagation constant $k_z$) is obtained by setting the determinant of this matrix to zero, determining the actual field amplitudes requires finding the numerical null space of this singular matrix.
  • Numerical Instability: The longitudinal field components ($E_z, H_z$), which are critical for chiral and vectorial light-matter interactions, are physically much smaller than the transverse components.
  • Consequence: Small numerical errors in the null-space calculation lead to disproportionately large relative errors in these crucial longitudinal components, compromising the accuracy of the full vectorial field structure.

2. Methodology

The authors propose a fundamentally different, semi-analytical approach that bypasses the ill-conditioned null-space calculation by exploiting the underlying symmetries of the cylindrical waveguide.

• Symmetry-Based Expansion: The method starts by expressing the fields using a mode expansion into eigenvectors that simultaneously diagonalize the symmetry operators of the waveguide: linear momentum ($\hat{p}_z$), angular momentum ($\hat{J}_z$), and energy ($\hat{H}$).
• Spin-Weighted Basis: The fields are expanded in a spin basis ($\hat{e}_0 = \hat{z}, \hat{e}_{\pm 1} = (\hat{x} \pm i\hat{y})/\sqrt{2}$), separating the problem into scalar radial dependencies multiplied by azimuthal and longitudinal phase factors.
• Strategic Normalization: A key innovation is the normalization of the radial dependence by the Bessel (or Hankel) function evaluated at the boundary ($\rho_0$). This allows the longitudinal field components at the boundary to be identified directly with two primary amplitudes, $A_{\ell m}$ (electric) and $B_{\ell m}$ (magnetic), rather than four.
• Analytical Reduction:
  1. By substituting this ansatz into Maxwell's equations, the transverse components are expressed analytically in terms of $A_{\ell m}$ and $B_{\ell m}$.
  2. Applying boundary conditions reduces the original $4 \times 4$ system to a $2 \times 2$ system of the form:
     $$\begin{pmatrix} \ell n_e & i\Phi^\mu_\ell \\ -i\Phi^\varepsilon_\ell & \ell n_e \end{pmatrix} \begin{pmatrix} A_{\ell m} \\ B_{\ell m} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$
     where $\Phi^\alpha_\ell$ are analytical functions involving Bessel/Hankel functions and material parameters.

3. Key Contributions

A. Reformulation to a $2 \times 2$ System

The paper demonstrates that the standard $4 \times 4$ problem is analytically equivalent to a $2 \times 2$ system. This reduction is the core novelty, offering two decisive advantages:

1. Robust Dispersion Relation: The dispersion relation is found by setting the determinant of the $2 \times 2$ matrix to zero, yielding a simple transcendental equation:
   $$\Phi^\varepsilon_\ell \Phi^\mu_\ell = (\ell n_e)^2$$
   This equation is well-behaved and ideal for numerical root-finding.
2. Analytical Amplitude Determination: Once the propagation constant ($k_z$) is found, the ratio between the electric and magnetic amplitudes ($\nu_{\ell m} = B_{\ell m}/A_{\ell m}$) is determined analytically from the remaining equations. This completely eliminates the need for numerical null-space calculations.

B. Enhanced Accuracy for Longitudinal Components

By calculating amplitudes analytically, the method preserves the accuracy of the small longitudinal field components ($E_z, H_z$). This is vital for applications relying on the precise polarization state and spin properties of the light (e.g., chiral quantum optics).

C. Generalized Mode Classification

The authors refine the naming convention for hybrid modes ($HE$ and $EH$). While standard definitions rely on the magnitude of longitudinal components ($|E_z|$ vs $|H_z|$), the authors propose a more general definition based on the relative phase between $E_z$ and $H_z$ (specifically $\text{Im}(AB^*)$), which remains valid even when material permeability $\mu_r \neq 1$.

D. Open-Source Implementation

The method is implemented in the open-source Python package Anafibre, which provides routines for:

• Solving the dispersion relation.
• Evaluating full vectorial fields (electric and magnetic).
• Normalizing modes to unit power using analytical integrals (avoiding numerical integration errors).

4. Results and Validation

• Numerical Stability: The paper demonstrates that the new method avoids the numerical divergence associated with the $4 \times 4$ null-space approach.
• Root Finding: The dispersion function $F_\ell(b, V)$ (where $b$ is the normalized propagation constant) is shown to be regular everywhere, with roots clearly identifiable as zero-crossings, making the search for modes robust.
• Analytical Integrals: The paper provides closed-form analytical expressions for the power normalization integrals (involving Bessel and Hankel functions), further reducing numerical noise and computational cost.
• Applicability: The method is general, handling arbitrary material parameters, including dispersive and lossy media (complex permittivity and permeability).

5. Significance

This work provides a powerful and reliable tool for the design and analysis of nanofibre-based devices.

• Scientific Impact: It resolves a long-standing subtle flaw in standard textbook methods, ensuring that simulations of nanofibre modes are physically accurate, particularly regarding the longitudinal fields that drive chiral interactions.
• Technological Impact: The method is essential for the next generation of nanophotonic applications, such as:
  • Chiral Quantum Optics: Precise control of atom-light interactions where the field polarization and longitudinal components dictate the coupling strength and directionality.
  • Vectorial Sensing: High-precision biosensing where the evanescent field's vectorial structure interacts with molecules.
  • Optical Trapping: Accurate modeling of optical forces on colloids and bacteria.

In summary, Golat and Rodríguez-Fortuño replace a numerically unstable, two-step procedure with a streamlined, semi-analytical framework that guarantees high fidelity in calculating the full vectorial electromagnetic modes of cylindrical nanofibres.