The Roulette Formalism

DRAFT under construction

TODO The formalism should be rewritten to use angular distances. In the implementation, the primary unit is pixels in the source plane. However, $\psi$ is a function of $\boldsymbol{\xi}$ in the lens plane, still measured by the same linear unit. It would be convenient if we could avoid the scaling and still get consistent nomenclature.

$The figure shows the set-up for the flat-sky approximation, with the source plane (the lens plane) a distance D_{\mathrm{S}} (D_{\mathrm{L}}) from the observer.$

The figure shows the set-up for the model used. In particular, the local coordinate systems used in the source plane and lens plane are shown.

\alpha\_s^m = - \frac{1}{2^{\delta\_{0s}}} D\_\textrm{L}^{m+1} \sum\_{k=0}^m\binom{m}{k} \left(\mathcal{C}\_s^{m(k)}\partial\_{\xi\_1}+\mathcal{C}\_s^{m(k+1)}\partial\_{\xi\_2}\right) \partial\_{\xi\_1}^{m-k}\partial\_{\xi\_2}^k\psi

(1)

\mathcal{C}\_s^{m(k)}=\frac{1}{\pi}\int\_{-\pi}^{\pi}{\rm d}\phi\sin^k\phi\cos^{m-k+1}\phi\cos s\phi

(2)

\beta\_s^m=-D\_\textrm{L}^{m+1}\sum\_{k=0}^m\binom{m}{k}\left({\mathcal{S}}\_s^{m(k)}\partial\_{\xi\_1}+{\mathcal{S}}\_s^{m(k+1)}\partial\_{\xi\_2}\right)\partial\_{\xi\_1}^{m-k}\partial\_{\xi\_2}^k\psi

(3)

\mathcal{S}\_s^{m(k)}=\frac{1}{\pi}\int\_{-\pi}^{\pi}{\rm d}\phi\sin^k\phi\cos^{m-k+1}\phi\sin s\phi.

(4)

The observed lensing is decomposed into two steps, as shown the figure. The first step is a translation (deflection), corresponding to the difference $\boldsymbol{\Delta\eta}$ between actual ( $\boldsymbol{\eta}\_\textrm{act}$ ) and apparent ( $\boldsymbol{\eta}\_\textrm{app}$ ) source-plane position. In the roulette formalism, this translational part of the lensing is given as $$

\boldsymbol{\Delta\eta} =\boldsymbol{\eta}\_\textrm{app}-\boldsymbol{\eta}\_\textrm{act} =-D\_\textrm{S}\cdot(\alpha^0\_1,\beta^0\_1),

(5)

$ $where$ (\alpha^0_1,\beta^0_1)$ is a vector of roulette amplitudes, as defined above.

The second step is the actual, non-linear distortion. The distorted image is drawn in a local co-ordinate system in the lens plane, centred at $\boldsymbol{\xi}=(\xi_1,\xi_2)$ , which corresponds to $\boldsymbol{\eta}\_\textrm{app}$ in the source plane. We write $\xi=|\boldsymbol{\xi}|$ for the distance between the distorted image and the lens in the lens plane. Since $\boldsymbol{\xi}$ and $\boldsymbol{\eta}\_{\mathrm{app}}$ lie on the same line through the viewpoint (cf. figure), we have

\xi = |\boldsymbol{\xi}| = \frac{D\_\textrm{L}}{D\_\textrm{S}}\cdot|\boldsymbol{\eta}\_{\mathrm{app}}|.

(6)

Following Clarkson, we use polar co-ordinates $(r,\phi)$ for the distorted image. The source image is described in Cartesian co-ordinates $(x^\prime,y^\prime)$ centered at $\boldsymbol{\eta}\_\textrm{act}$ in the source plane. Thus the light observed at a position (pixel) $(r,\phi)$ is drawn from a different position (pixel) $(x',y')=\mathcal{D}$ $(r,\phi)$ in the source image. From~Eq.~48 in \citet{Clarkson_2016_II} it is possible to show that the mapping $\mathcal{D}$ is given as $$ \begin{aligned} \frac{D_{\mathrm{L}}}{D_{\mathrm{S}}}\cdot

\begin{bmatrix} x' \\\\ y' \end{bmatrix}

(7)

r\cdot\begin{bmatrix} \cos\phi \\ \sin\phi \end{bmatrix} + \sum_{m=1}^{\infty} \frac{r^m}{m!\cdot D_{\mathrm{L}}^{m-1}} \cdot\sum_{s=0}^{m+1} c_{m+s} \left(\alpha_s^m \boldsymbol{A}_{s} + \beta_s^m \boldsymbol{B}_{s} \right) \begin{bmatrix} C^+ \\ C^- \end{bmatrix} \\\\ C^\pm &= \pm \frac{s}{m+1},\\\\ c_{m+s} &= \frac{1 - (-1)^{m+s}}{4} = \begin{cases} 0, \quad m+s \text{ is even},\\\\ \frac12, \quad m+s \text{ is odd}, \end{cases} \\\\ \boldsymbol{A}_{s} &= \begin{bmatrix} \cos{(s-1)\phi} & \cos{(s+1)\phi} \\\\ -\sin{(s-1)\phi} & \sin{(s+1)\phi} \end{bmatrix}, \\\\ \boldsymbol{B}_{s} &= \begin{bmatrix} \sin{(s-1)\phi} & \sin{(s+1)\phi} \\\\ \cos{(s-1)\phi} & -\cos{(s+1)\phi} \end{bmatrix}. \end{aligned} $ $The coefficients$ \alpha_m^s $and$ \beta_m^s $depend on the lens potential$ \psi(\xi_1,\xi_2)$, from which one may derive the physical properties of the lens.

In practice the sum has to be truncated by limiting $m\le m_0$ for some $m_0$ .

The roulette amplitudes are always calculated in a specific Reference Point.