[Paper Review] Some Aspects of the Electromagnetic Multipole Expansions

Irina Dumitriu, C. Vrejoiu|ArXiv.org|Mar 19, 2005

Quantum and Classical Electrodynamics3 references3 citations

TL;DR

This paper presents a systematic Cartesian tensor approach to electromagnetic multipole expansions, emphasizing algebraic reduction to symmetric traceless tensors for calculating radiated power. It highlights the emergence of toroidal multipole contributions and provides a nontrivial grouping of multipolar terms, offering a computationally simpler alternative to spherical harmonic methods with explicit expressions for power radiated up to high-order multipoles including corrections via time derivatives of moments.

ABSTRACT

Various procedures for expressing the multipolar expansion of the electromagnetic field are considered with application to the calculation of the radiated power. Some results from literature are discussed and perspective of developing the subject is pointed out.

Motivation & Objective

• To develop a systematic and algebraic method for electromagnetic multipole expansions in Cartesian coordinates, avoiding special functions.
• To extend the reduction of Cartesian multipole tensors to symmetric traceless forms beyond the static case into the dynamic regime.
• To clarify and compare contributions from electric, magnetic, and toroidal multipoles in the total radiated power, particularly highlighting toroidal moments.
• To provide a consistent framework for computing radiated power up to high-order multipoles using time-derivative corrections of moments.
• To demonstrate the advantages of Cartesian formalism in simplifying calculations and revealing hidden multipolar structures such as toroidal contributions.

Proposed method

• Uses Cartesian coordinate components for multipole tensors, avoiding spherical harmonics and special functions.
• Applies a generalized tensor reduction technique (from Ref. [5]) to convert arbitrary Cartesian tensors into fully symmetric, traceless forms.
• Derives the radiated power expression using reduced multipole moments, including time derivatives of moments up to fourth order.
• Introduces and defines the electric toroid moment via a recursive algorithm based on higher-order multipole tensors.
• Applies the reduction scheme to both charge and current density expansions, ensuring gauge invariance and consistency.
• Uses a systematic notation for multipole moments (e.g., $\mathbf{P}^{(n)}$, $\mathbf{M}^{(n)}$, $\mathbf{T}^{(n)}$) and their time derivatives to express power contributions.

Experimental results

Research questions

• RQ1 How can the electromagnetic multipole expansion be systematically formulated in Cartesian coordinates without relying on spherical harmonics?
• RQ2 What is the role of toroidal multipole moments in the total radiated power, and how do they emerge from the tensor reduction procedure?
• RQ3 How do time derivatives of multipole moments (e.g., $\ddot{\Lambda}^{(n)}$) contribute to the radiated power at high orders?
• RQ4 What is the relationship between the reduced symmetric traceless tensors and the standard multipole moments in the literature?
• RQ5 In what way does the Cartesian formalism simplify the calculation of radiated power compared to traditional spherical harmonic methods?

Key findings

• The Cartesian approach simplifies the formalism by replacing special functions with algebraic manipulations and combinatorics.
• The reduction procedure reveals nontrivial groupings of multipolar contributions, particularly isolating the electric toroid moment $\mathbf{T}^{(n)}$ as a distinct physical entity.
• The radiated power expression includes corrections from higher-order time derivatives of moments, such as $\frac{1}{c^2}\dot{\mathbf{T}}^{(1)}$ and $\frac{1}{c^4}\ddot{\Lambda}[\mathbf{N}^{(4,1)}_{\text{sym}}]$, up to order $\mathcal{O}(1/c^4)$.
• The electric toroid moment is defined as $\mathbf{T}^{(n)} = \frac{n}{(n+1)^2}\widetilde{\mathbf{N}}^{(n+1,1)} - \frac{n}{2(n+2)}\dot{\widetilde{\Pi}}^{(n)}$, showing its origin in the reduction of higher-order multipole tensors.
• The method successfully reproduces and clarifies results from earlier works (e.g., [6,7]) and compares favorably with other literature methods, especially in identifying contributions from $\mathcal{M}^{(2)}$, $\mathcal{P}^{(3)}$, and $\widetilde{\mathbf{N}}^{(4,1)}$.
• The formalism consistently accounts for multipole contributions up to fifth order, with explicit expressions for $\mathcal{P}^{(5)}$, $\mathcal{M}^{(4)}$, and $\widetilde{\mathbf{N}}^{(4,3)}$ in the power formula.