Multiple Imaging in the Weak-Field Limit

Now that we have had some practice in carrying out lensing calculations and have derived the all-important factor of two, we turn to non-axisymmetric lens models in the context of the thin lens approximation. We show in Sect. 4.1 that the vector counterpart of the deflection angle is given as the gradient of a scalar potential and uses Fermat’s principle to obtain the lens equation. We then solve this equation to find the image positions for a simple but important special case. What follows thereafter is a general discussion of image magnification (Sect. 4.2) and time delay (Sect. 4.3), which can be observed and compared against specified lens models. Some general properties of the lens mapping are described in Sects. 4.4 and 4.5. We derive the conservation of surface brightness in Sect. 4.6, which allows us to consider lensing of spatially extended sources. Mathematical degeneracies in the lens equation, which constrain our ability to compare theory and observation, are discussed in Sect. 4.7. The case of a light ray deflected by multiple lenses (Sect. 4.8) rounds out our presentation of the theory of strong lensing in the weak-field limit.