3. Multiplane lensing recursion
Planes are sorted by increasing redshift. A ray observed at image-plane angle $\boldsymbol{\theta}$ (a 2-D vector in arcsec) is traced forward through each plane in order. Its angular position at plane $j$ is given by the multiplane recursion (Schneider, Ehlers & Falco 1992):
Multiplane recursion
\(\boldsymbol{\theta}_j \;=\; \boldsymbol{\theta} \;-\; \sum_{k\,<\,j} \frac{D_{kj}}{D_j}\;\hat{\boldsymbol{\alpha}}_k\left(\boldsymbol{\theta}_k\right)\)
| Symbol | Meaning |
|---|---|
| $\boldsymbol{\theta}$ | Observed angle (image-plane position); fixed for each rendered pixel |
| $\boldsymbol{\theta}_j$ | Ray’s angular position at plane $j$ (arcsec) |
| $D_{kj}$ | Angular diameter distance from plane $k$ to plane $j$ |
| $D_j$ | Angular diameter distance from observer to plane $j$ |
| $\hat{\boldsymbol{\alpha}}_k(\boldsymbol{\theta}_k)$ | Deflection angle from all lens objects in plane $k$, evaluated at the ray’s position $\boldsymbol{\theta}_k$ |
Each object carries its own type, lens or source, independently of which plane it belongs to. Only lens objects enter the deflection sum; source objects are passive and receive the ray without contributing to it. A plane containing both types is a hybrid plane; its lens objects deflect and its source objects emit, handled separately by the same recursion. The weight $D_{kj}/D_j$ converts the deflection at plane $k$ into its angular displacement at the later plane $j$.
The position at the target plane is the source-plane position $\boldsymbol{\beta}$, where source brightness is sampled.
Key idea. Because each lens plane evaluates its deflection at the ray’s already-deflected position $\boldsymbol{\theta}_k$, successive lens planes interact non-linearly, a key feature of multiplane lensing absent in single-plane calculations.
