# Lenses

## SIS

The `sis`

lens is a singular isothermal sphere with deflection^{1}
\[
\alpha_x = r_E \, \frac{x}{r} \;,
\]
\[
\alpha_y = r_E \, \frac{y}{r} \;,
\]
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 deflection^{1}
\[
\alpha_x = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctan} \left( \frac{x \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2}} \right) \;,
\]
\[
\alpha_y = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctanh} \left( \frac{y \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2}} \right)
\]

### 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 deflection^{1}
\[
\alpha_x = r_E \, \frac{x}{r + s} \;,
\]
\[
\alpha_y = r_E \, \frac{y}{r + s}
\]

### 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 deflection^{1}
\[
\alpha_x = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctan} \left( \frac{x \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2} + s} \right) \;,
\]
\[
\alpha_y = r_E \, \frac{\sqrt{q}}{\sqrt{1 - q^2}} \, \text{arctanh} \left( \frac{y \, \sqrt{1 - q^2}}{\sqrt{q^2 x^2 + y^2} + q^2 s} \right)
\]

### 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.