# Lenses

## SIS

The sis lens is a singular isothermal sphere with deflection1 $\alpha_x = r_E \, \frac{x}{r} \;,$ $\alpha_y = r_E \, \frac{y}{r} \;,$ where $r_E$ is the Einstein radius, and $r$ is the distance to the position of the lens.

## SIE

The sie lens is a singular isothermal ellipsoid with deflection1 $\alpha_x = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctan} \left( \frac{x \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2}} \right) \;,$ $\alpha_y = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctanh} \left( \frac{y \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2}} \right)$

### Notes

When the axis ratio q is fixed to unity, the lens becomes a singular isothermal sphere, but the implemented deflection diverges. Use the sis lens in this case.

## NSIS

The nsis lens is a non-singular isothermal sphere with deflection1 $\alpha_x = r_E \, \frac{x}{r + s} \;,$ $\alpha_y = r_E \, \frac{y}{r + s}$

### Notes

When the core radius s is fixed to zero, the lens becomes a singular isothermal sphere. Use the sis lens in this case.

## NSIE

The nsie lens is a non-singular isothermal ellipsoid with deflection1 $\alpha_x = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctan} \left( \frac{x \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2} + s} \right) \;,$ $\alpha_y = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctanh} \left( \frac{y \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2} + q^2 s} \right)$

### Notes

When the axis ratio q is fixed to unity, the lens becomes a non-singular isothermal sphere, but the implemented deflection diverges. Use the nsis lens in this case.

When the core radius s is fixed to zero, the lens becomes a singular isothermal ellipsoid. Use the sie lens in this case.

## EPL

The epl lens follows an elliptical power law profile 2

$\kappa(R) = \frac{2-t}{2} \left(\frac{b}{R}\right)^t$

where $R$ is the elliptical radius $R = \sqrt{q^2 x^2 + y^2}$, $b$ is the scale length, and $t$ is the slope of the power law.

### Notes

When the axis ratio $q$ is fixed to unity, the lens becomes a regular power law lens.

When the slope $t$ is fixed to unity, the lens becomes a singular isothermal ellipsoid. Use the sie lens in this case.

When the slope $t$ is fixed to 2, the lens becomes a point mass. Use the point_mass lens in this case.

1. P. Schneider, C. S. Kochanek, and J. Wambsganss, Gravitational Lensing: Strong, Weak and Micro (Springer, 2006).

2. N. Tessore & R. B. Metcalf, A&A (2015).