The angular diameter distance quoted in Equation 1 is only useful
in a homogeneous universe. Inhomogeneous universes necessitate
drastic approximations. Most treatments of this problem assume that
the universe is uniform on the scale of the Hubble radius and that the
relationship between redshift and cosmological time is the same as
that in a FRW universe of similar mean density.
We are interested in the evolution of the cross section of a null
geodesic congruence (a bundle of rays) as it propagates backward
in time away from the observer,
specifically in the mapping from the observer's angle
space to proper distance perpendicular to some central or
fiducial ray. Let us generalize
Figure 5
and set the vector,
()=(_{1}, _{2}), equal
to this offset as measured all
along the bundle, not just in the lens plane. The subscripts 1, 2
represent components with respect to an orthonormal basis parallel
propagated along the fiducial ray, and is the affine
parameter. Next, define the complex number
() = _{1} +
i _{2}. The angle between a ray in the
congruence and the fiducial ray at the observer can also be
represented as a complex number dot_{0}, where a dot denotes
differentiation with respect to . We are now in a position
to generalize the notion of angular diameter distance by defining a
two component vector, D =
(D_{1}, D_{2}), where both
D_{1} and D_{2} are complex, using the
general linear relation
9.
The real part of D_{1} measures the expansion of the ray
while its imaginary part describes pure rotation. (In practice,
rotation is usually small and D_{1} is approximately real.)
D_{2} measures the shear. All the information about the local
image distortion is contained in D. The conventional angular
diameter distance, whose square is the ratio of the source area to the
solid angle it subtends, is given by [ D_{1} ^{ 2} 
 D_{2} ^{ 2}] ^{1/2}, and suffices for point
sources where only the flux can be measured.
In an inhomogeneous universe containing Newtonian matter, D
can be shown to evolve according to
10.
where the quantity R = (1 + z)^{2}(_{,11} + _{,22}) =
4(1 +
z)^{2}G describes focusing by matter lying within the
congruence with pr oper density , and F = (1 + z)^{2}(_{,11}  _{,22} + 2i _{,12}) describes the
influence of matter external to the congruence (e.g.
Penrose 1966,
Blandford et
al. 1991).
This formalism immediately gives expressions for the
magnification tensors, [µ] (cf Equation 3), whose definition we
can now generalize by identifying with the angle which
would be subtended by the proper length in the source plane in a
FRW universe of similar average density to the inhomogeneous
universe under consideration. (See
Ehlers &
Schneider 1986
for an alternative choice of reference universe).
In the limiting case when all the matter in the universe apart from
the lens is isolated from the congruence (D_{2} = 0), the lack of
focusing by matter in the beam (save for the lens) compared to a FRW
universe of the same _{0} increases the angular diameter
distance of the source
(Dashevskii &
Zel'dovich 1965,
Dyer & Roeder 1972,
Nottale 1983,
Nottale & Hammer
1984,
Kasai et
al. 1990).
The increase is about 30 per cent for a source
with z_{s} ~ 2 in an EinsteinDe Sitter universe.
However, the cumulative
shear caused by external matter usually produces a second order
focusing which leads to a diminished net effect. In general, if
multiple imaging is uncommon, the distribution of magnifications due
to smoothly distributed matter is dominated by the convergence rather
than shear
(Lee & Paczynski
1990,
Watanabe &
Sasaki 1990).
The total flux is always conserved when suitably averaged over all
directions
(Weinberg 1976,
Peacock 1986).
If the focusing (say due to a lensing galaxy) is strong enough to make
the rays cross along any congruence (Figure 8),
then multiple images
must form and we have an example of gravitational lensing. At the
point where the rays cross, known as a conjugate point to the
observer, the conventional angular diameter distance vanishes (
D_{1}  =  D_{2} ) and the formal magnification
diverges. The locus
of these conjugate points is a twodimensional surface, a caustic
sheet, to which the rays are tangent (see
Blandford &
Narayan 1986
for a schematic diagram showing the caustic sheets
associated with an elliptical lens). Equivalently, we can think in
terms of wavefronts normal to the rays merging at a caustic
(Kayser &
Refsdal 1983).
For a source at a fixed redshift, the source plane
intersects the caustic sheets at caustic lines. The images of these
lines are known as critical curves (cf
Figures 6,
7).

Figure 8. An infinitesimal conical bundle of rays is shown
drawn backwards from an observer, past an elliptical lens, and
touching two caustic sheets. The second caustic sheet, on the left,
has a cusp line perpendicular to the plane of the diagram, while the
first caustic sheet has a cusp in the plane. Representative cross
sections of the bundle are indicated at the bottom. Where the bundle
touches a caustic, its cross section degenerates to a straight
line. Beyond this point, the bundle is ``inverted'' and a source
located here will acquire two additional images. In general there
could be many caustic sheets behind a complex lens, but with a single
elliptical lens there are only two sheets (which may penetrate each
other, cf Blandford & Narayan 1986).

In the generic situation, the caustic sheet corresponds to a fold
caustic. When a source crosses a fold, an extra pair of images
will either be created or destroyed. These image pairs will be
stretched toward each other along a direction essentially
perpendicular to the projection of the caustic on the sky
(Blandford &
Kovner 1988).
Because of the stretching, the images will be
bright; an example is the pair of bright images,
A_{1}A_{2}, in
Q1115+080. The magnifications of the two images will
be inversely
proportional to their separations and also inversely proportional to
the square root of the distance of the source from the caustic
(Benson & Cooke
1979,
Ohanian 1983,
Blandford &
Narayan 1986,
Kayser & Witt
1989).
Therefore, for a fold caustic, the cross section,
( > µ), for the magnification to be greater than
µ has a universal scaling, µ^{2},
for µ >> 1.
Equivalently, the differential cross section scales as d / dµ
µ^{3}.
Every time a ray touches a caustic (grazes it tangentially), the
associated image is inverted, i.e. its parity is reversed.
(Polarization directions are parallel propagated and unaffected.) In
Q0957+561, the A image is believed to be uninverted
while the ray
associated with the B image has touched one caustic, so the two images
are approximate reflections of each other. A faint third image ought
to be formed in the galaxy nucleus, inverted twice through roughly
orthogonal planes, hence rotated through ~ 180°.
Fold surfaces meet at cusp lines, which correspond to a cusp
caustic. Sources lying just inside cusps create three bright images
(plus any additional images that are not associated with the cusp).
Sources lying just outside cusps have one of their images highly
brightened. In this region, the cross section for large µ
scales as
(> µ)
µ^{5/2}, or d /dµ µ^{7/2}. Cusps are believed to play
an important role in the
luminous arcs. Cusp lines meet at points associated with higher
order singularities, but these have not yet been identified in the
observations. The closest point of the caustic to the observer is
generically a cusp. When a source is located close to this point,
the lens is said to be marginal and may produce one or three
bright images
(Narayan et
al. 1984,
Kovner 1987d,
e).
In general, for a non singular lens, caustic surfaces separate regions
with image multiplicities differing by two. Since far from the lens a
source has but one image, therefore the total number of images has to be
odd
(Burke 1981,
McKenzie 1985).