SPM is a spherical refracting surface separating two media of refractive indices n1 and n2...

SPM is a spherical refracting surface separating two media of refractive indices n1 and n2. (see Fig. 3.37). Consider an object point O forming a virtual image at the point I. We assume that all rays emanating from O appear to emanate from I so as to form a perfect image. Thus according to Fermat’s principle, we must have

n1 OS– n2 SI= n1 OP– n2 PI

where S is an arbitrary point on the refracting surface. Assuming the right hand side to be zero, show that the refracting surface is spherical, with the radius given by

    (95)

Thus show that

    (96)

where d1 and d2 are defined in Fig. 3.37; (see also sec. 4.10).

[Hint: We consider a point C which is at a distance d1 from the point O and d2 from the point I. Assume the origin to be at O and let (x, y, z) represent the coordinates of the point S. Thus

 n1 (x2 + y2 + z2) ½ – n2 (x2 + y2 + Δ2) ½ = n1 (r + d1) – n2 (r + d2) = 0

where Δ = d2 – d1. The above equation would give the equation of a sphere whose center is at a distance of n2r/n1 (= d1) from O.]