384 Gravitational lensing
Figure 14.2.Notation for the lens geometry.14.2.3 Lens equation
The geometry for a typical gravitational lens is given in figure 14.2 A light
ray from a source S (inη) is deflected by the lens by an angleα(with impact
parameter|ξ|) and reaches the observer located in O.
The angle between the optical axis (arbitrarily defined) and the true source
position is given byβ, whereas the angle between the optical axis and the image
position isθ. The distances between the observer and the lens, the lens and the
source, and the observer and the source are, respectively,Dd,DdsandDs.From
figure 14.2 one can easily derive (assuming small angles) thatθDs=βDs+αDds.
Thus the positions of the source and the image are related by the following
equation:
β=θ−α(θ)Dds
Ds, (14.13)
which is called thelens equation. It is a nonlinear equation so that it is possible
to have several imagesθcorresponding to a single source positionβ.
The lens equation (14.13) can also be derived using the Fermat principle,
which is identical to the classical one in geometrical optics but with the refraction
index defined as in equation (14.1). The light trajectory is then given by the
variational principle
δ∫
ndl= 0. (14.14)It expresses the fact that the light trajectory will be such that the travelling time
will be extremal. Let us consider a light ray emitted from the source S at time