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I want to simulate gravitational lensing. In the article LENSINGGW: a PYTHON package for lensing of gravitational waves there is this fragment [page 3]:

enter image description here

My research: As far as I understand, this is a simplified formulation. Here $\kappa(\boldsymbol{\theta})$ - is the distribution of mass in the plane of the lens. $\psi(\boldsymbol{\theta})$ - two dimensional deflection potential. $\boldsymbol{\theta},\boldsymbol{\beta},\boldsymbol{\xi},\boldsymbol{\eta}$ - certain coordinates that characterize the position of the source image and its already lensed image (which, obviously, is in the plane of the lens).

How I imagine the solution algorithm using these formulas:

  1. First we define the distribution of masses in the plane of the lens $\kappa(\boldsymbol{\theta})$.
  2. From equation (3) we find the deflection potential.
  3. Substitute deflection potential into formula (2), replace $\boldsymbol{\beta}$ with $\boldsymbol{\eta}/D_{S}$ (as follows from formula (1) on page 2).
  4. We solve the equations for $\boldsymbol{\theta}$ and multiply it by $D_L$, which also follows from equation (1) and obtain the coordinates of the source in the plane of the gravitational lens $\boldsymbol{\xi}$.

My question is: I don't fully understand the components of this equation. Are coordinates $\boldsymbol{\theta},\boldsymbol{\beta},\boldsymbol{\xi},\boldsymbol{\eta}$ Cartesian coordinates or are they some kind of angular coordinates? How to solve the 2D Poisson equation (3) for a given mass distribution (probably only a numerical solution is possible)? What exactly do the coordinates $\boldsymbol{\theta},\boldsymbol{\beta},\boldsymbol{\xi},\boldsymbol{\eta}$ characterize - wave amplitude, intensity, or something else? Is it true that the conversion of source coordinates to lensed coordinates is carried out directly through equation (2)?

I would appreciate and appreciate for help in understanding these issues.

Remark: I first asked this question on Mathematica Stack Exchange because I wanted to understand these formulas and at the same time write code for the lens there. Soon I came across another code that was more convenient to remake for Matlab. But one way or another, in all the algorithms I found there are points that are not entirely clear to me. Therefore, if anyone would like to leave a comment or even an answer, you are welcome!

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