Proceeding from Fermat'sprinciple derive the refraction formula for paraxial rays on a spherical boundary surface of radius `R` between media with refractive indices `n` and `n'`.

Explanation

For `O_(1)` to be image, the optical paths of all rays `OAO_(1)` must be equal upto terms of leading order in `h`. Thus
`n_(1)OA + n_(2) AO_(1) =` constant
But, `OP = |s|, O_(1)P = |s|` and so
`OA = sqrt(h^(2) + (|s| + del)^(2)) ~= |s| + del + (h^(2))/(2|s|)`
`O_(1)A = sqrt(h^(2) + (|s'| - del)^(2)) ~= |s'| - del + (h^(2))/(2|s'|)`
(neglecting products `h^(2) del)`. Then
`n_(1) |s| + n_(2) |s'| + n_(1) del - n_(2) del + (h^(2))/(2) ((n_(1))/(|s|)+(n_(2))/(|s'|)) =` const.
Now `(r - del)^(2) + h^(2) = r^(2)`
or `h^(2) = 2r del` or `del = (h^(2))/(2r)`
Here `r = CP.`
Hence `n_(1) |s| + n_(2) |s'| + (h^(2))/(2) {(n_(1) -n_(2))/(r ) + (n_(1))/(|s|) + (n_(2))/(s'|)} =` constant
Since this must hold for all `h`, we have
`(n_(2))/(|s'|) + (n_(1))/(|s|) = (n_(2) - n_(1))/(r )`
From our sign convention, `s' gt 0, s lt 0` so we get
`(n_(2))/(s') - (n_(1))/(s) = (n_(2) - n_(1))/(r )`.