(i) Consider a thin lens placed between a source (S) and an observer (O), Fig. Let the thickness of the lens vary as `w (b) = w_0 - (b^2)/(prop)`, where `b` is the vertical distance from the pole. `w_0` is a constant. Using Fermat's principle, i.e., the time of transit for a ray between the source and observer is an extremism, find the condition that al 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) = k_1 In ((k_2)/(b)) b_min lt b lt b_(max) w(b) = k_1 In ((k_2)/(b_(min))) b lt 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 `beta = sqrt(((n - 1) k_1 (u)/(v))/(u + v))`.
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Explanation

The time elapsed to travel from S to `P_(1)` is
`" " t_(1)=(SP_(1))/(c)=sqrt(u^(2)+b^(2))/(c)`
`or " " (u)/(c) (1+(1)/(2)(b^(2))/(u^(2)))` assuming `b lt lt u_(0)`
The time required to travel from `P_(1)` to O is
`" " t_(2)=(P_(1)O)/(c)=sqrt(v^(2)+b^(2))/(c),(1+(1)/(2)(b^(2))/(v^(2)))`
The time required to travel through the lens is `" " t_(1)=((n-1)w(b))/(c)`
where n is the refractive index.
Thus, the total time is
`" " t=(1)/(c)u+v+(1)/(2)b^(2)((1)/(u)+(1)/(v))+(n-1)w(b)`
Put `" " (1)/(D)=(1)/(u)+(1)/(v)`
Then, `" " t=(t)/(c)(u+v+(1)/(2)(b^(2))/(D)+(n-1)(w_(0)+(b^(2))/(alpha)))`
Fermet's principle gives the time taken should be minimum.
For that first derivative should be zero
`" " (dt)/(db)=0=(b)/(CD)-(2(n-1)b)/(calpha)`
`" " alpha=2(n-1)D`
Thus, a convergent lens is formed if `alpha=2 (n-1) D.` This is independent of and hence, all paraxial rays from S will converge at O i.e., for rays
and `" " (b lt lt v.)`
Since , `(1)/(D)=(1)/(u)+(1)/(v)`, the focal length is D.
(ii) In this case, differentiating expression of time takem t w.r.t. b
`" " t=(1)/(c)(u+v+(1)/(2)(b^(2))/(D)+(n-1) k_(1)ln((k_(2))/(b)))`
`" " (dt)/(db)=0 =(b)/(D)-(n-1)(k_(1))/(b)`
`rArr" " b^(2)=(n-1)k_(1)D`
`:. " " b=sqrt((n-1)k_(1)D)`
Thus, all rays passing at a height b shall contribute to the image. The ray paths make an angle.
`" " beta,(b)/(v)=(sqrt((n-1)k_(1))D)/(v^(2))=sqrt(((n-1)k_(1)uv)/(v^(2)(u+v)))=sqrt(((n-1)k_(1)u)/((u+v)v))`
This is the required expression.