MultiPlane

When the deflectors along the line of sight are located at different redshifts (e.g., a galaxy lens with line-of-sight perturbers, or a cluster with structures at multiple redshifts), the single-plane approximation breaks down, and the light rays need to be propagated through a series of lens planes.

Following same terminology as single plane lensing, we can write the lensing potential and deflection in $j$th plane as,

\[\begin{align*} ψ_j(\pmb{θ}_j) &= \frac{4{\rm G}}{\rm c^2} \frac{1}{D_j} \int d^2 \pmb{θ}' \, Σ_j(\pmb{θ}') \, \ln|\pmb{θ}_j - \pmb{θ'}|, \\ \pmb{α}_j(\pmb{θ}_j) &= \pmb{∇}_j ψ_j(\pmb{θ}_j), \end{align*}\]

where $D_j$ is the angular diameter distance from observer to the $j$th lens plane and $Σ_j$ is the surface mass density in the $j$th lens plane. Starting from observer, by the time a ray reaches $j$th lens plane it has already deflected in previous $j-1$ lens planes. Hence, the impact parameter in the 1st lens plane is related to impact parameters in $j$th lens plane as,

\[\pmb{θ}_j = \pmb{θ}_1 - \sum_{i=1}^{j-1} \frac{D_{ij}}{D_j} \pmb{α}_i(\pmb{θ}_i).\]

With the above, the total potential and deflection for a lens system with $N_p$ planes is given as,

\[\begin{align*} ψ(\pmb{θ}) &= \sum_{i=1}^{N_p} \frac{D_{is}}{D_s} ψ_i(\pmb{θ}_i), \\ \pmb{α}(\pmb{θ}) &= \sum_{i=1}^{N_p} \frac{D_{is}}{D_s} \pmb{α}_i(\pmb{θ}_i). \end{align*}\]

Similarly, the excess time delay accumulates between consecutive planes as,

\[\begin{equation*} Δt = \sum_{i=1}^{N_p} \frac{1 + z_i}{\rm c} \frac{D_i D_{i+1}}{D_{i,i+1}} \left[ \frac{1}{2} \left| \pmb{θ}_i - \pmb{θ}_{i+1} \right|^2 - \frac{D_{i,i+1}}{D_{i+1}} ψ_i(\pmb{θ}_i) \right], \end{equation*}\]

where $z_i$ is the redshift of the $i$th lens plane and the ($N_p$+1)th plane is the source plane, i.e., $\pmb{θ}_{N_p+1} = \pmb{β}$ and $D_{N_p+1} = D_s$.

In LensFactory, a multi-plane lens is initialized from a vector of lens components, where each component specifies its lens redshift (z_d) in addition to the usual lens model parameters. Components sharing the same redshift are automatically grouped into a single (composite) lens plane, and the lens planes are sorted in increasing redshift.

LensFactory.Lenses.init_MultiPlaneLens — Type

init_MultiPlaneLens(lens::Vector{<:NamedTuple};
                    cosmology::Union{Nothing, Cosmology.AbstractCosmology} = nothing,
                    position::Symbol = :physical)

Initialize a multi-plane lens from a vector of lens components. Each component must contain the lens redshift (z_d) along with the parameters of the corresponding lens model. Components sharing the same redshift are grouped into a single (composite) lens plane, and the lens planes are sorted in increasing redshift. At least two distinct lens planes are required.

Arguments

lens::Vector{<:NamedTuple}: Vector of lens components.

Keyword Arguments

cosmology = nothing: Cosmology model. Required only when position = :observed to calculate the plane-plane distance ratios.
position = :physical: Convention for the supplied deflector centres.
- :physical: centres are already plane-frame positions and are used as-is.
- :observed: centres are observed (on-sky) positions. Those on planes behind the first need to be mapped to their physical (plane-frame) positions by ray-tracing through the foreground planes.

Returns

MultiPlaneLens: Multi-plane lens.