Question

i) Consider a thin lens placed between a source (S) and an observer (O) (Fig. 9.8). Let the thickness of the lens vary as w(b)=w_o-b²/α , where b is the verticle distance from the pole. w₀ is a constant. Using Fermat’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 O on the axis. Find the focal length.

(ii) A gravitational lens may be assumed to have a varying width of the form

w(b)=k₁ln(k₂/b)

b_min<b<b_max

=k₁ln(k₂/b_min)

b<b_min

Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius

β=√(((n-1)k₁u/v)/u+v)

Answer 1

Thus a convergent lens is formed if α = 2(n – 1)D. This is independant of b and hence all paraxial rays from S will converge at O (i.e. for rays b << n and b << v).

Thus all rays passing at a height b shall contribute to the image. The ray paths make an angle