Differenting SIE APproach 1

This is one of several approaches suggested for differentiation of the lens potential for SIE.

Approach 1. Rotation of the Coordinate System¶

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

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

This gives