Optics, electromagnetic theory Definition

Discrete dipole approximation

Builds on Electric dipole moment Electromagnetic scattering Clausius–Mossotti relation

Definition

Let a scatterer be illuminated by a time-harmonic incident electric field $E_{\mathrm{inc}}$ of angular frequency $\omega$ . Choose $N$ points $r_1, \dots, r_N$ that fill the scatterer, usually on a cubic lattice of spacing $d$ much smaller than the wavelength. Give each point $r_j$ a polarizability $\alpha_j$ , a complex $3 \times 3$ matrix or scalar. For $j \ne l$ , let $G_{jl}$ be a $3 \times 3$ matrix with the following property. For each $p \in \C^3$ , $G_{jl} p$ is the electric field at $r_j$ radiated by an oscillating electric dipole of moment $p$ at $r_l$ . The discrete dipole approximation takes the dipole moments $p_1, \dots, p_N \in \C^3$ to solve $p_j = \alpha_j \Big( E_{\mathrm{inc}}(r_j) + \sum_{l \ne j} G_{jl} \, p_l \Big), \qquad j = 1, \dots, N.$ It takes the scattered field to be the sum of the fields radiated by these $N$ dipoles.

Remarks

The system has $3N$ complex unknowns. The polarizabilities are chosen so that the array mimics the material: a basic choice is the Clausius–Mossotti relation with a small radiative correction.

Example: for $N = 1$ , a small sphere is one dipole with $p_1 = \alpha_1 E_{\mathrm{inc}}(r_1)$ , and the scattered field is the field of that dipole. This recovers Rayleigh scattering by a small sphere.

Non-example: taking $p_j = \alpha_j E_{\mathrm{inc}}(r_j)$ for all $j$ when $N > 1$ drops the coupling terms $G_{jl} p_l$ . This is a first-order Born-type approximation, not the discrete dipole approximation.

The method converges to the exact solution as $d \to 0$ for fixed size and refractive index. It needs $\abs{m} k d$ small, where $m$ is the refractive index of the material and $k$ the wavenumber in vacuum. The large linear system is solved iteratively, with fast Fourier transforms for the lattice sums.

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Electric dipole moment concept
Electric field concept
Electromagnetic scattering concept
Permittivity concept
Polarization density concept
Clausius–Mossotti relation statement
Discrete dipole approximation definition

Tag	`BDHTRF8RMF`
URI	`m2://BDHTRF8RMF`
Kind	definition
Writing	W0 stub
MSC	78A45, 78M99
Defines	Discrete dipole approximation
Uses	Electric dipole moment, Electromagnetic scattering, Clausius–Mossotti relation
Changed	3 November 2026

Discrete dipole approximation

Definition

Remarks

Reading path

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