The Roulette Formalism

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

The observed lensing is decomposed into two steps, as shown by the figure. The first step is a translation (deflection), corresponding to the difference between apparent ( $\boldsymbol{\theta}$ ) and actual ( $\boldsymbol{\beta}$ ) source position:

\boldsymbol{\alpha}=\boldsymbol{\theta}-\boldsymbol{\beta}.

(1)

We have already seen that the deflection angle $\boldsymbol{\alpha}$ may be expressed as the gradient of the two-dimensional projected lensing potential. In the roulette formalism this translates into a set of so-called roulette amplitudes. In particular

\boldsymbol{\alpha}=\nabla_\theta\psi^\textrm{R}=-(\alpha^0_1,\beta^0_1),

(2)

where $(\alpha_1^0,\beta_1^0)$ is a vector of roulette amplitudes, as defined above.

The second step is the actual, non-linear distortion, and may be expresed through higher-order derivatives of the lensing potential. The distorted image is drawn in a local co-ordinate system in the lens plane, centred at $\boldsymbol{\theta}=(\theta_1,\theta_2)$ . Following Clarkson, we use polar co-ordinates for the distorted image. While Clarkson used $(r,\phi)$ we will use $(\tilde{\theta},\phi)$ , since we use angular units.

The source image is described in Cartesian co-ordinates $(\tilde{\beta}_1,\tilde{\beta}_2)^\textrm{T}$ centered at $\boldsymbol{\beta}$ in the source plane.

The lens plane, viewed face-on. The dashed vector
\boldsymbol{\theta}=(\theta_1,\theta_2) gives the position of the
image’s light centre (the small dot) relative to the lens. The local
polar co-ordinate system (\tilde{\theta},\phi) is centred on this
light centre, with \tilde\theta the (angular) radius and \phi the
angle from the local polar axis (parallel to the global \theta_1
direction). The global coordinate directions \theta_1,\theta_2,
centred on the lens, are shown for reference. — Figure 2:The lens plane, viewed face-on. The dashed vector $\boldsymbol{\theta}=(\theta_1,\theta_2)$ gives the position of the image’s light centre (the small dot) relative to the lens. The local polar co-ordinate system $(\tilde{\theta},\phi)$ is centred on this light centre, with $\tilde\theta$ the (angular) radius and $\phi$ the angle from the local polar axis (parallel to the global $\theta_1$ direction). The global coordinate directions $\theta_1,\theta_2$ , centred on the lens, are shown for reference.

The source plane, viewed face-on. The dashed vector
\boldsymbol{\beta}=(\beta_1,\beta_2) gives the position of the
source’s light centre (the small dot) relative to the optical axis.
The local Cartesian co-ordinate system (\tilde{\beta}_1,\tilde{\beta}_2)
is centred on this light centre, parallel to the global
\beta_1,\beta_2 axes. The global coordinate directions
\beta_1,\beta_2, centred on the optical axis, are shown for
reference. — Figure 3:The source plane, viewed face-on. The dashed vector $\boldsymbol{\beta}=(\beta_1,\beta_2)$ gives the position of the source’s light centre (the small dot) relative to the optical axis. The local Cartesian co-ordinate system $(\tilde{\beta}_1,\tilde{\beta}_2)$ is centred on this light centre, parallel to the global $\beta_1,\beta_2$ axes. The global coordinate directions $\beta_1,\beta_2$ , centred on the optical axis, are shown for reference.

Thus the light observed at a position (pixel) $(\tilde{\theta},\phi)$ is drawn from a different position (pixel) $(\tilde{\beta}_1,\tilde{\beta}_2)=\mathcal{D}(\tilde{\theta},\phi)$ in the source image. From Eq. (48) in Clarkson (2016) it is possible to show that the mapping $\mathcal{D}$ is given as

\begin{aligned} \begin{bmatrix} \tilde{\beta}_1 \\\\ \tilde{\beta}_2 \end{bmatrix} &= \tilde{\theta}\cdot\begin{bmatrix} \cos\phi \\\\ \sin\phi \end{bmatrix} + \sum_{m=1}^{\infty} \frac{\tilde{\theta}^m}{m!} \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}

(3)

The coefficients $\alpha_m^s$ and $\beta_m^s$ depend on the lens potential $\psi^\textrm{R}(\theta_1,\theta_2)$ , 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. In the next section we will give expressions for calculating the roulette amoplitudes.

References¶

Clarkson, C. (2016). Roulettes: a weak lensing formalism for strong lensing: II. Derivation and analysis. Classical and Quantum Gravity, 33(24), 245003. 10.1088/0264-9381/33/24/245003