Superpotential-Based Cylindrical Vector Wave Functions

• SUPER-CVWFs are defined as vector field representations derived from scalar superpotentials satisfying the Helmholtz equation, offering a complete modal basis.
• They employ gradient, curl, and double-curl operators on separable scalar solutions to generate orthogonal wave functions for isotropic and gyrotropic media.
• This framework enhances practical modal analysis in cylindrical domains, supporting accurate computational methods in electromagnetic and elastodynamic research.

Superpotential-Based Cylindrical Vector Wave Functions (SUPER-CVWFs) are an advanced framework for representing vector fields in cylindrical geometries, built upon scalar “superpotentials” that satisfy the scalar Helmholtz equation. They yield a complete, orthogonal basis for expanding general vector solutions of system PDEs—such as Maxwell's equations—in homogeneous isotropic and gyrotropic media, and form the core representation in contemporary modal analysis for open waveguides and scattering problems (Eom et al., 2017, Delimaris et al., 4 Jan 2026). SUPER-CVWFs offer a mathematically rigorous, symmetry-adapted set of vector wave functions, crucial for both theoretical analysis and computational implementation in electromagnetic and elastodynamic settings.

1. Definition and Mathematical Structure

SUPER-CVWFs are constructed by combining scalar solutions—superpotentials—to the Helmholtz equation in cylindrical coordinates. For homogeneous isotropic media, three scalar superpotentials $\phi_{i}(r,\theta,z)$ ($i=1,2,3$) are each separable as

$$\phi_{i,m}(r,\theta,z)=A_{i,m} J_m(k_{r,i} r) e^{im\theta} e^{ik_z z}$$

where $J_m$ is the Bessel function of order $m$, $k_{r,i}^2 = k_i^2 - k_z^2$, and $k_z$ is the axial wavenumber.

From these, three types of vector wave functions are defined:

• Type-I: $E_{m,k_z}^{(1)} = \nabla\times [e_z\,\phi_{1,m}]$
• Type-II: $$E_{m,k_z}^{(2)} = \nabla\times\nabla\times [e_z\,\phi_{2,m}]$$
• Type-III: $E_{m,k_z}^{(3)} = \nabla[\phi_{3,m}]$

Explicit expressions in cylindrical coordinates are derived for each component, encompassing all necessary axisymmetric and helical field structures (Eom et al., 2017).

In gyrotropic (anisotropic, magneto-optic) media, the framework generalizes using constitutive tensors,

$$\boldsymbol\epsilon = \epsilon_0 \begin{pmatrix} \epsilon & -i\epsilon_a & 0 \ i\epsilon_a & \epsilon & 0 \ 0 & 0 & \epsilon_z \end{pmatrix} ,\quad \boldsymbol\mu = \mu_0 \begin{pmatrix} \mu & -i\mu_a & 0 \ i\mu_a & \mu & 0 \ 0 & 0 & \mu_z \end{pmatrix}$$

Field solutions are expanded in vector wave functions derived from two scalar superpotentials $\Phi_j(\rho,\phi)$, each satisfying

$(\nabla_T^2 + \chi_j^2)\Phi_j = 0$

with the transverse wavenumbers $\chi_j$ given by roots of a quartic dispersion relation (Delimaris et al., 4 Jan 2026).

2. Vector Wave Function Construction

The vector wave functions in both isotropic and gyrotropic cases are obtained by action of differential operators (gradient, curl, double curl) on superpotentials aligned along coordinate axes. For the gyrotropic case, the electric and magnetic fields inside the core are expanded as

$$\mathbf{E}(\rho,\phi,z) = \sum_{j=1}^2\sum_{m=-\infty}^{\infty} A_{jm} \Big[\tilde{\mathbf M}_m^{(1)}(\chi_j) + \tilde{\mathbf N}_m^{(1)}(\chi_j) + \tilde{\mathbf L}_m^{(1)}(\chi_j)\Big] e^{i\beta z}$$

where the vector functions $\tilde{\mathbf M}_m^{(1)}$, $\tilde{\mathbf N}_m^{(1)}$, and $\tilde{\mathbf L}_m^{(1)}$ are combinations of Bessel functions and tensor-prefactors determined by the underlying constitutive relations.

In the cladding—assumed isotropic—the expansion uses decaying cylindrical vector wave functions with modified Bessel functions, and appropriate matching of normalization to dyadic Green’s function solutions (Delimaris et al., 4 Jan 2026).

3. Orthogonality, Normalization, and Completeness

SUPER-CVWFs are orthogonal under volume or surface bilinear forms, such as

$$\int_V E_{m, k_z}^{(i)} \cdot \bigl[E_{m',k_z'}^{(j)}\bigr]^*\, dV = N_{m}^{(i)} \delta_{i,j} \delta_{m,m'} \delta(k_z - k_z') \frac{\delta(k_{r,i} - k_{r,j}')}{k_{r,i}}$$

and, for gyrotropic cases, the surface form

$$\langle(\mathbf E, \mathbf H), (\mathbf E', \mathbf H')\rangle = \tfrac12 \int_S \left[\mathbf E \times \mathbf H'^{*} + \mathbf E'^{*}\times \mathbf H\right]\cdot\hat{\mathbf e}_z\,dS$$

Normalization constants may be chosen so that the basis is orthonormal ($N_m^{(i)}=1$).

The completeness property holds: any admissible vector field can be expanded as a (possibly infinite) sum/integral over the corresponding indices, with expansion coefficients determined by orthogonality integrals. The completeness relation in dyadic form ensures that the SUPER-CVWFs form a resolution of identity in the relevant Hilbert space (Eom et al., 2017, Delimaris et al., 4 Jan 2026).

4. Application to Boundary Value Problems in Cylindrical Domains

SUPER-CVWFs provide a canonical modal basis for solving boundary value problems in cylindrical domains where partial differential equations of mathematical physics—e.g., Maxwell's equations, elastodynamic equations—permit separation of variables in cylindrical coordinates.

In both isotropic and gyrotropic settings, an arbitrary field configuration (for example, the electromagnetic field in a fiber or a gyromagnetic rod) is represented as a sum over the modal indices; boundary conditions at interfaces (such as core-cladding boundaries) are enforced by projecting onto this basis. For gyrotropic waveguides, matching tangential field components leads to a homogeneous linear system for the modal amplitudes, with the propagation constants $\beta$ determined by the roots of determinant conditions (Delimaris et al., 4 Jan 2026).

Two notable numerical approaches are employed:

• Extended Integral Equation (EIE) Method: Boundary conditions are enforced via integral representations involving free-space dyadic Green’s functions, leading to matrix systems whose nontrivial solutions yield propagation constants.
• Chebyshev Expansion Method (CEM): Boundary conditions are projected onto Chebyshev polynomial bases sampled at Chebyshev nodes on the boundary, resulting in matrix eigenvalue problems for $\beta$.

Both methods rely on the expressiveness and completeness of the SUPER-CVWF basis in the relevant geometry.

5. Modal Classification: TE, TM, and Hybrid Modes

In isotropic media, the modal structure decouples into pure TE and TM sets. In gyrotropic media, the coupling between field components arising from the off-diagonal elements ($\epsilon_a$, $\mu_a$) leads generically to hybrid modes with both $E_z$ and $H_z$ nonzero. Only in limiting cases—where either $\chi_j\rightarrow 0$ or specific coupling coefficients vanish—do pure TE or TM modes emerge.

Each $(j,m)$ pair corresponds in general to a hybrid mode branch; the radial order $p$ arises from roots of the characteristic equation for $\beta$. For isotropic limits ($\epsilon_a = \mu_a = 0$), the two superpotential indices $j=1,2$ recover the conventional TE$_m$ and TM$_m$ modal families (Delimaris et al., 4 Jan 2026).

6. Contemporary Applications and Computational Modal Analysis

SUPER-CVWFs have become a standard tool for modal analysis in advanced open gyrotropic waveguides, including non-circular geometries and magneto-optic cores. Their construction enables full-wave solutions “without approximations”, supporting accurate computation of propagation constants and modal fields for both lossless and lossy materials. Benchmark studies have confirmed agreement between SUPER-CVWF-based solvers and commercial finite element packages (Delimaris et al., 4 Jan 2026).

Practical applications include the analysis and design of gyroelectric and gyromagnetic waveguides subject to external bias (e.g., ferrite rods in microwave engineering), as well as canonical problems in viscoelasticity and elastodynamics (Eom et al., 2017).

7. Significance and Theoretical Impact

The introduction of superpotential-based CVWFs provides a mathematically rigorous, constructively complete, and computationally efficient basis for modal decomposition in cylindrical geometries, unifying prior approaches for isotropic and anisotropic (gyrotropic) media. The ability to handle oblique incidence, arbitrary external fields, and non-circular boundaries extends the modal toolbox to a broad class of modern electrodynamic and elastodynamic systems.

A plausible implication is the facilitation of further generalizations to higher-rank tensor media and the systematic study of boundary-value problems with complex spatial symmetries. Their adoption in recent modal analysis—such as in (Delimaris et al., 4 Jan 2026)—suggests continued centrality in the analysis of waveguide and scattering phenomena with cylindrical (or near-cylindrical) symmetry.