CosmoSim is a simulator for gravitational lensing.
This is one of several approaches suggested for differentiation of the lens potential for SIE.
The deflection is given as the vector $\nabla\psi^{\mathrm{R}}$, in a Cartesian co-ordinate system with axes aligned with the axes of the lens. We call this the lens frame. We are interested in the deflection vector $\vec{\alpha}(x,y)$ in a global frame, which shares the origin with the lens frame, but is rotated clockwise by an angle $\theta$. In other words, the lens is oriented at an angle $\theta$ (counterclockwise) in the global frame.
We will let $(x,y)$ denote the point in the global frame, and
$(x’,y’)$ the same point in the lens frame. Hence
\begin{aligned}
\begin{bmatrix} x’ \\\\ y’ \end{bmatrix}
&=
\begin{bmatrix}
\cos\theta & \sin\theta \\\
-\sin\theta & \cos\theta
\end{bmatrix}
\cdot
\begin{bmatrix} x \\\\ y \end{bmatrix}
\end{aligned}
In other words, the $(x,y)$ coordinates are rotated clockwise.
Similarly the deflection is given as $\nabla\psi^{\mathrm{R}}(x’,y’)$ in
the lens frame, and $\vec{\alpha}(x,y)$ in the global frame.
Thus, $\nabla\psi^{\mathrm{R}}$ has to be rotated counterclockwise, as
\begin{aligned}
\vec{\alpha}(x,y)
&=
\begin{bmatrix}
\cos\theta & -\sin\theta \\\
\sin\theta & \cos\theta
\end{bmatrix}
\cdot
\nabla\psi^{\mathrm{R}}(x’,y’)
\end{aligned}
This gives
\begin{aligned}
\vec{\alpha}(x,y) =
C_0\frac{\sqrt{f}}{f’}\cdot
\begin{bmatrix}
\cos\theta\cdot\sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x\cos\theta+y\sin\theta}{R}\right)
-\sin\theta\cdot\sin^{-1}\left(f’\cdot\frac{-x\sin\theta+y\cos\theta}{R}\right)
\\\
\sin\theta\cdot\sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x\cos\theta+y\sin\theta}{R}\right)
+\cos\theta\cdot\sin^{-1}\left(f’\cdot\frac{-x\sin\theta+y\cos\theta}{R}\right)
\end{bmatrix}
\end{aligned}