Implementation of SIE

Raytrace¶

This is implemented in the SIE class (src/simlib/SIE.cpp).

Evaluation of the lens potential.¶

To evaluate the lens potential $\psi^{\mathrm{R}}(x,y)$ , we calculate the polar coordinates $R$ and $\phi$ , and use the formula (see SIE) for $\psi_{\xi_0,f,\theta,D_\mathrm{L}}^\textrm{SIE(R)}(R,\phi)$ .

Evaluation of the Deflection.¶

This uses \begin{aligned} \frac{\partial\psi^{\mathrm{R}}}{\partial x} &= C_0\frac{\sqrt{f}}{f’}\cdot \left( \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) \right) \\\\ \frac{\partial\psi^{\mathrm{R}}}{\partial y} &= C_0\frac{\sqrt{f}}{f’}\cdot \left( \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) \right) \end{aligned} as taken from the expression for $\vec\alpha(x,y)$ above.

Note in polar co-ordinates, we replace $x/R$ and $y/R$ with $\cos\phi$ and $\sin\phi$ respectively, and find that $\nabla\psi$ is constant in $R$ .

Roulette¶

Under Construction

Step 1. [Differentiation of SIE](Differentiation of SIE)

\begin{align} \alpha^m_s & = \Gamma^m_s \square^{a^-} \sum_{k=0}^{2k\le s} (-1)^k \binom{s}{2k} \frac{\partial^{s}}{(\partial x)^{s-2k}(\partial y)^{2k}} \\\\\\\\ \beta^m_s & = \Gamma^m_s \square^{a^-} \sum_{k=0}^{2k+1\le s} (-1)^k \binom{s}{2k+1} \frac{\partial^{s}}{(\partial x)^{s-2k-1}(\partial y)^{2k+1}} \end{align}

(1)

where

\Gamma^m_s = \begin{cases} -(2^{-\delta_{0s}})\frac{\chi^{m+1}}{2^m}\binom{m+1}{(m+1-s)/2}, \quad\text{$m+s$ odd} \\\\\\\\ 0 \quad\text{$m+s$ even} \end{cases}

(2)

and

\square = (\nabla\psi)^2

(3)

TODO $a^-$

Deprecated notes
- [Roulette Amplitudes in SIE](Roulette Amplitudes in SIE)