), Fig. 6(EP).18. Let the thickness
of the lens vary as w (b) 
,where b is the
vertical distance from the pole. 
is a " constant
Using Format's principle, i.e., the time of transit for a ray between the
source and observer is an extremum, find the condition that all paraxial rays
starting from the source will converge at a point on
the axis. Find the focal length.
(ii) A
gravitational lens may be assumed to have a varying width of the form
Show that
an observe will see an image of a point object as a ring about the center of
the lens with an angular radius 
Answer:
Refer to
Fig.6(EP).18. Time required by light to travel in air from S to P1 is
,
Assuming b<<u
Similarly, time required
by light to travel in air from P1 to 
Now, thickness of
lens varies as 
?(i)
Time required to travel through the
lens is
Here, n is
refractive index of the material of the lens.
Thus, the total time
of travel from S to O is
Let us put 
?(ii)
?(iii)
According to
Fermat?s principle, t = extremum. Therefore 
.
From (iii),
or 
?(iv)
Hence, a convergent lens is formed if . This is independent of b. Hence,
all paraxial rays from S will converge at 
From (ii), .Therefore,
focal length of lens =D.
(ii) Now, for a gravitational lens,
Proceeding
as in the above case, from eqn.(iii)
Hence, all rays
passing through the lens at a height b shall form the image. The paths of rays
would make an angle
Hence, the observe will see an image of a point object as a
ring about the centre of the lens with an angular radius
.
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