Basic Notation

we assume a flat sky and a thin lens, which means the lens is concentrated in one plane, orthogonal on the line of sight, with no extension in depth. We call this the lens plane $L$. Similarly, the source that we observe through the lens, is concentrated in a parallel plane, called the source plane $S$. The distances to the lens and source planes are denoted $D_L$ and $D_S$ respectively. The optical axis is the line from the observer through the lens.

We consider a single source point at $\boldsymbol{\eta}$ in $S$. The apparent position, as seen by the observer, is at $\nu$ and

The apparent position in the lens plane $L$ is called

The deflection is most easily described in terms of angles, so we define $\beta$ and $\theta$ as the angles between the optical axis and respectively $\boldsymbol{\eta}$ and $\boldsymbol{\nu}$. The deflection angle $\hat\alpha$ is the angle between the actual and apparent source in the source plane as seen from the apparent image in the lens plane.

With the flat sky approximation, the angles are related to lengths in the lens plane by a factor of $D_L$, so that

Similarly, in the source plane^[1], we find

but also

Now, we can write the actual image as

It is noteworthy that if we consider spherical geometries (a source sphere and a lens sphere), we may also write down the equation

Considering small angles,

it may readily be shown that Eq.~(6) and Eq.~(7) are the same. It is this latter equation that we shall take to be our constitutive relation. But before we get there, let us also introduce the standard way of normalizing.

Normalisation¶

It is customary to normalise using a constant factor $\xi_0$. This gives the following entities, following Kormann 1994

In addition to the previously mentioned (cosmological) distances $D_L$, $D_S$ and $D_{LS}$ we must thus find a proper length scale $\xi_0$ from which we normalize everything else. In SEF, Kormann1994 and other standard sources one typically takes $\xi_0$ to be the so-called Einstein radius. This is the radius at which a spherically symmetric lens will produce a ring (so-called Einstein ring) whenever the source is directly behind the lens, along the optical axis.

In normalised coordinates the ray-trace equation reads

which also explains the particular definition of $\boldsymbol{\alpha}$: it makes the normalized version of the ray-trace equation look very nice and tidy. The normalisation presented above is however somewhat different from the one we shall prefer in this work, where we shall prefer to work in angular coordinates, such as the ones given in Eq.~(7). In the sections to follow, such coordinates will therefore be our focus.

The lens-potential $\psi$¶

Considering a thin lens, it is customary to define the lens potential as the projection of the 3D gravitaitonal lens potential down on the lens plane. Such a simplification is typically warranted, due to $D_L \gg \xi_0$.

To connect with SEF and other standard literature we will start by defining the lensing potential so that its gradient is the so-called reduced deflection angle $\mathbf{a}$ the gradient of $\psi$, i.e.

In angular coordintes, which we prefer here, the chain rule gives^[2],

The normalised raytrace equation~(10) is thus rewritten to

Working on the sphere we also had gives

This final equation shall be taken as motivation to redefine the standard lensing potential ever so slightly. Let $\psi$ be the usual lensing potential. Then we define

In this notation, we find the following pleasing expression for the angular version of the ray-trace equation:

It follows that

Polar and Cartesian Coordinates¶

Refering to two-dimensional points we will need both Cartesian and Polar co-ordinates.
For the apparent position $\theta$ we will write $(\theta_1,\theta_2)$ for the Cartesian co-ordinates, and $(\theta,\phi)$ for the polar co-ordinates. For $\beta$ we will write $(\beta_1,\beta_2)$ and $(\beta,\phi_\beta)$ respectively.