(i) The time required to travel from S to P1 is t1 = SP1/c= \(\sqrt {u^2+b^2}\over{c}\); u/c (1 + 1/2 \(\frac{b^2}{u^2}\)) assuming b <u0
The time required to travel from P1 to O is
The time required to travel through the lens is t1 = \(\frac{(n-1)w(b)}{c}\) where is the refractive index.
Thus, the total time is
Fermet’s principles gives
\(\frac{dt}{db}=∩=\frac{b}{cd}\frac{2(n-1)b}{cα}\)
α = 2(n-1)D
Thus, a convergent lens is formed if α = 2(n – 1) D. This is independent of b and hence all paraxial rays from S will converge at O (i.e. for rays b < < n and b < < v).
Since \(\frac{1}{D}=\frac{1}{u}+\frac{1}{v}\), the focal length is D.
(ii) In this case
Thus, all rays passing at a height b shall contribute to the image. The ray paths make an angle