# Far field cross-sections

### Extinction

The extinction cross-section may be obtained from the optical theorem as the interference between incident and scattered fields,

$$$\sigma_\text{ext} = 4\pi\kappa^{-1} k_1 \Im\left(\mathbf{P} \cdot \mathbf{E_\text{inc}}^* \right).$$$
CoupledDipole.extinction!Function
extinction(kn::Real, P::Array{Complex}, Ein::Array{Complex})

Extinction cross-section for each incident angle

• kn: wavenumber in incident medium
• P: 3N_dip x N_inc matrix, polarisations for all incidences
• Ein: 3N_dip x N_inc matrix, incident field for all incidences
source

### Absorption

The absorption cross-section is obtained by evaluating the work done by the total field, (but excluding self-reaction), on the dipoles:

$$$\sigma_\text{abs} = 4\pi \kappa^{-1} k_1 \left[\Im\left(\mathbf{P} \cdot \mathbf{E}^* \right) - \frac 2 3 \kappa^{-1} k_1^3 |\mathbf{P}|^2\right].$$$
CoupledDipole.absorption!Function
absorption(kn::Real, P::Array{Complex}, E::Array{Complex})

Absorption cross-section for each incident angle

• kn: wavenumber in incident medium
• P: 3N_dip x N_inc matrix, polarisations for all incidences
• E: 3N_dip x N_inc matrix, total field for all incidences
source

### Scattering

The scattering cross-section can be computed in two ways:

• as the difference between extinction and absorption
• by integrating the flux of the Poynting vector over all scattering directions in the far-field.

Comparing both results might be useful as a consistency check.

$$$\sigma_\text{sca} = \kappa^{-2} k_1^4 \iint_\Omega \left|\sum_i \left(\mathbb{I} - \mathbf{\hat n}\otimes\mathbf{\hat n}\right) \mathbf{P}_i \mathrm{exp}(-ik_1 \mathbf{r}_i\cdot\mathbf{\hat n})\right|^2 \text{d}\Omega.$$$
CoupledDipole.scattering!Function
scattering(positions::Vector{SVector{3}}, ScatteringVectors::Vector{SVector{3}}, weights::Vector{Real}, kn::Real, P::Array{Complex})

Scattering cross-section for each incident angle, obtained by numerical cubature over the full solid angle of scattering directions

• positions: vector of cluster particle positions
• ScatteringProjectorz: N_inc-vector of far-field directions
• weights: N_inc-vector of cubature weights
• kn: wavenumber in incident medium
• P: 3N_dip x N_inc matrix, polarisations for all incidences
source