Consider the idealised system shown in Figure 2.1, where the symbols D_{s}, D_{l} and D_{ls} are used for the distance from the observer to the background source, the observer to the lens and the lens to the source respectively.

All of these deflection angles are very small. Therefore we can use an important result from geometry; for an isosceles triangle (one with two sides of equal length) with two very long sides and one short side, the product of the small angle between the long sides and the length of one of the long sides is equal to the length of the short side, provided that the angle is expressed in radians. Luckily this is already the case in the equation we saw earlier for the deflection angle in a gravitational field.
You can therefore see that the length OQ is just D_{s}, and OQ' is D_{s}. [If you're worrying that these triangles are not isosceles but rather are rightangled then you can think about the isosceles triangles which would be formed by reflecting these rightangled triangles in the horizontal plane  the angles at the apex of these triangles are then 2 and 2 but the lengths at the bases are also doubled so the same formulae result. Alternatively if you are happy with trigonometry you could consider the tangent of the angles and remember that tangents of small angles are equal to the angle itself measured in radians.]
Now consider the triangle PQQ'; you can derive from this triangle that the distance QQ' is D_{ls}.
Obviously OQ + QQ' = OQ', and so
D_{s} + D_{ls} = D_{s}
so rearranging, we finally get
= (D_{s}/D_{ls})  (D_{s}/D_{ls})
which is known as the "lens equation".
If we were to plot against we would therefore get a straightline graph with gradient (D_{s}/D_{ls}) and intercept  (D_{s}/D_{ls}) .
So far we have not included any information about the physics of exactly how the light gets deflected. We can do this quite easily if we remember for a point mass lens that
= 4GM/bc^{2}
and the distance of closest approach, b, is just D_{l}; therefore = (4GM/c^{2}) x 1/(D_{l})  in other words, the deflection angle decreases for rays which pass a greater distance from the centre of the lens.
Obviously the point mass is not a particularly
good representation of a real galaxy. Real galaxy "lenses" have a
much slower falloff of deflection angle with distance, and in fact a constant
(call it _{0})
which just reverses sign as the light path goes one side or the other from the
galaxy centre is a good approximation. This is known as the
"isothermal" model, and corresponds to a surface mass density which
falls off as 1/r, where r is the distance from the centre of the galaxy. The
abrupt change in corresponds to a singularity in the mass
distribution in the centre, and such galaxies are known as "singular
isothermal ellipsoids", or "singular isothermal spheres" for
those that are spherical in distribution rather than elliptical.
For an image to be formed, we must demand that
both equations  the lens equation and our idea of the dependence of and  are
satisfied simultaneously. You should convince yourself that in this simple
model, either one or three images of the background source can be formed,
depending on how close the lines of sight are to the lensing object.
In practice, more complicated analysis in two
dimensions allows the possibility of five lensed images, provided that the
lines of sight to background source and lens are close enough. The gravitational lens simulator available on the web from
Jodrell Bank allows you to investigate the types of images that are seen for
different positions of the source compared to the lens centre. Look especially
for the following, see Figure 2.2:

Use
the
simulator to investigate the
dependence of the images' appearance on the parameters in the simulation. In
particular, notice the following: (i) How does the galaxy ellipticity change
the types of image seen as you move the source closer to the galaxy? (ii)
What effect does the Einstein radius have on the simulation? 
Individual images within the gravitational lens system form, as we have seen, with different magnification. Some of the images also have different parity, a quantity which can be positive or negative. A positiveparity image of a Zshaped source would appear as a Zshaped image (although possibly distorted), whereas a negativeparity image would appear Sshaped. In general, negativeparity images form closer to the centre of the lens. For example, in a twoimage system the fainter image closer to the centre of the lens has negative parity, and the brighter, more distant image has positive parity.
There is a second, rather elegant way to examine
the geometry of a gravitational lens system. It is by the use of Fermat's
principle, originally stated by the French physicist Pierre
de Fermat (16081665). In the form he originally stated it, it reads:
The actual path between two points taken by a
beam of light is the one which is traversed in the least time. 
The more modern formulation, which is needed in studies of gravitational lensing, reads:
The actual path between two points taken by a
beam of light is the one which is an extremum  that is a maximum, a minimum or a saddle
point. 
A saddle point is strictly speaking neither a maximum or minimum, but as its name implies is a locally flat area similar to that at the centre of a saddle  it resembles a maximum in one direction (across the horse) and a minimum in another (along the horse). In all three cases the main point is that the difference in light travel time which results from going a small distance from the extremum is vanishingly small. In the language of calculus, this means that the first derivative of the path with respect to small deviations vanishes.
For light propagating through a uniform medium from point A to point B Fermat's principle is very simple. The light ray will just follow the path of minimum time  in other words, a straight line.
For light propagating through a gravitational lens, the situation is different. Suppose that the source, the lens and the observer are all in a straight line. If the light now follows a "straight line" it has to go through the centre of the lens, which contains a large gravitational field. The effect of this field is to retard the light as it struggles out, and more time is taken. The shortest path would therefore be one which avoids the centre of the lens and is deflected to reach the observer, but not so far that a large overhead is incurred by travelling a longer distance. The resulting compromise corresponds exactly to the light path which we derived earlier using the lens equation. In practice, not only the minimumtime path will be followed but, due to the complicated nature of the lens, images also form which have followed maximum paths and paths corresponding to saddle points.
Try the gravitational lens
simulator again, but this time selecting the option to "Plot Fermat
surfaces". These purple lines are contours of constant light travel
time; in other words, light from the source which has gone through any
point on each connected line will take the same time to reach the observer.
Repeat the previous lens simulation exercise, but this time note that:
Jim
Lovell (U. Tasmania, 
Any concentrated mass can in principle act as a gravitational lens. Whether multiple images are visible depends on the concentration of mass at the centre of the lens, the central surface mass density (a surface density is the mass per unit area, normal densities are volume densities i.e. mass per unit volume). Only above a critical value of central surface mass density (CSMD) will the characteristic multiple imaging be seen.
Table 1 shows the three major types of lensing which are astronomically important (adapted from a similar table in a review article by Refsdal & Surdej). In most of this course we will consider only lensing by single galaxies. However, lensing by stars ("microlensing") and lensing by clusters of galaxies will be discussed in section 7.
Lens 
Lensed object 
Central surface mass density 
Einstein radius 
Single galaxy 
Usually quasars; can be background galaxies 
For normal lensing galaxies, a few times critical CSMD 
About 1 arcsecond 
Single star (or massive object, e.g. black hole) in our galaxy 
Background star in our galaxy or Magellanic Clouds 
Vastly greater than critical CSMD (lensing object very compact) 
About 1 milliarcsecond (very hard to resolve) 
Cluster of galaxies 
Many background galaxies 
About critical CSMD in cluster centre, stretching and distortion also seen 
Tens of arcseconds 
As was derived above , the angular radius (in
radians) of an Einstein Ring is given by
^{2}=4GMD_{ls}
/ c^{2}D_{l}D_{s} .
Use this formula to confirm the orders of magnitude of the Einstein radii presented in Table 1. You will need to choose appropriate values for the various distances and lens masses.
In the next section we will discuss the way in which we search for lenses.
1. Using the equations given so far, work out an equation for the angular size for the Einstein ring produced by a point mass M in terms of the distances to lens and source. You will need the lens equation, remembering that for the Einstein ring =0, and Einstein's equation for light deflection by a point mass.
Answer to question
Angular size is given by ^{2}=4GMD_{ls} / c^{2}D_{l}D_{s}.
From the geometry (fig 2.1), b=D_{l}, so the equation for a point source becomes =4GM/D_{l}c^{2}. Combining this with the rearranged lens equation: =D_{s}/D_{ls} gives the result.
2. Try the same exercise for a galaxy, modelling it instead as a singular isothermal sphere.
Answer to question
=D_{ls}_{0}/D_{s}
This
one is easier: the singular isothermal model has a constant deflection, which
we call _{0}.
Substituting this into the lens equation (with =0) gives
the required result.