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.