[Paper Review] Spectral decomposition of the Lippmann-Schwinger equation applied to cylinders

Parry Y. Chen, David J. Bergman+1 more|arXiv (Cornell University)|May 4, 2017

Experimental and Theoretical Physics Studies4 citations

TL;DR

This paper presents a spectral decomposition method for solving the Lippmann-Schwinger equation in cylindrical structures by leveraging eigenmodes of the system to analytically compute electromagnetic fields. By expressing the total field as a superposition of eigenmodes, the method enables rapid, analytic evaluation of scattering for any source configuration without repeated numerical solves.

ABSTRACT

We derive the spectral decomposition of the Lippmann-Schwinger equation for electrodynamics, obtaining the fields as a sum of eigenmodes. The method is applied to cylindrical geometries.

Motivation & Objective

• To develop an analytic framework for solving the Lippmann-Schwinger equation in cylindrical geometries with arbitrary source configurations.
• To overcome the computational burden of iterative numerical solvers by using eigenmodes as a basis for field decomposition.
• To unify the treatment of free sources and structural response via a common Green’s function, enabling analytical solutions.
• To enable rapid computation of electromagnetic fields for any source position, orientation, or extent using a single precomputed eigenmode basis.

Proposed method

• Formulate the Lippmann-Schwinger equation using a unified Green’s function that applies to both free sources and structural inhomogeneities.
• Decompose the problem into two parts: the radiation pattern of free sources in a background medium and the eigenmodes of the structure.
• Expand the total field in terms of cylindrical harmonic eigenmodes, which are solutions to the homogeneous Lippmann-Schwinger equation.
• Use the orthogonality and completeness of the eigenmodes to project the source field and compute the coupling coefficients analytically.
• Derive explicit expressions for the eigenmodes in cylindrical coordinates using Bessel functions and vector spherical harmonics.
• Evaluate the normalization and coupling integrals using analytical expressions involving Bessel functions and their integrals.

Experimental results

Research questions

• RQ1 Can the Lippmann-Schwinger equation for cylindrical structures be solved analytically using eigenmode decomposition?
• RQ2 How can the response of a dielectric cylinder to arbitrary external sources be computed without repeated numerical simulations?
• RQ3 What are the analytical forms of the eigenmodes of the Lippmann-Schwinger equation in cylindrical geometry?
• RQ4 How does the unified Green’s function simplify the treatment of both free sources and structural inhomogeneities in scattering problems?
• RQ5 What is the role of the eigenmode basis in enabling rapid, analytic evaluation of field responses for any source configuration?

Key findings

• The eigenmodes of the Lippmann-Schwinger equation in cylindrical geometry are analytically expressed using Bessel functions and vector harmonics, with explicit forms for transverse electric and magnetic components.
• The normalization factor for the eigenmodes is derived as a combination of Bessel function integrals, yielding a closed-form expression involving $ I_m(eta, eta, a) $.
• The coupling between the source field and the eigenmodes is computed analytically, enabling the total field to be reconstructed as a superposition of eigenmodes with known coefficients.
• The method allows for the analytic evaluation of the field response to any source configuration (position, orientation, extent) once the eigenmodes are computed.
• The approach avoids iterative numerical solvers and provides a direct, non-iterative solution to the Lippmann-Schwinger equation in cylindrical systems.
• The derived eigenmode basis is complete and orthogonal, ensuring accurate and stable field reconstruction across the entire domain.