Thin-lens approximation for gravitational lensing ($\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }(\mathbf{\theta })$)

The gravitational lensing approximation in which deflection is concentrated in a plane whose thickness is small relative to the relevant observer-lens-source distances. The reduced angular deflection includes the distance ratio. A projected surface mass density then determines the lensing potential and the image-to-source mapping. This is distinct from a refracting thin optical lens.