All Problems Photometry and Geometrical Optics
Problem 5.29
Proceeding from Fermat's principle derive the refraction formula for paraxial rays on a spherical boundary surface of radius \(R\) between media with refractive indices \(n\) and \(n^{\prime}\).
5.29. Suppose \(S\) is a point source of light and \(S^{\prime}\) its image (Fig. 38). According to Fermat's principle the optical paths of all rays originating at \(S\) and converging at \(S^{\prime}\) are equal. Let us draw circles with the centres at \(S\) and \(S^{\prime}\) and radii \(S O\) and \(S^{\prime} M .\) Consequently, the optical paths \((D M)\) and \((O B)\) must be equal: \(n \cdot D M=n^{\prime} \cdot O B\) However, in the case of paraxial rays \(D M \approx A O+O C,\) where \(A O \approx h^{2} /(-2 s)\) and \(O C \approx h^{\prime 2} / 2 R .\) Besides, \(O B=O C-B C \approx\) \(\approx h^{\prime 2} / 2 R-h^{\prime 2} / 2 s^{\prime} .\) Substituting these expressions into \((*)\) and taking into account that \(h^{\prime} \approx h,\) we obtain \(n^{\prime} / s^{\prime}-n / s=\left(n^{\prime}-n\right) / R\).