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 RR between media with refractive indices nn and n′n^{\prime}.

Reveal Answer
5.29. Suppose SS is a point source of light and S′S^{\prime} its image (Fig. 38). According to Fermat's principle the optical paths of all rays originating at SS and converging at S′S^{\prime} are equal. Let us draw circles with the centres at SS and S′S^{\prime} and radii SOS O and S′M.S^{\prime} M . Consequently, the optical paths (DM)(D M) and (OB)(O B) must be equal: n⋅DM=n′⋅OBn \cdot D M=n^{\prime} \cdot O B However, in the case of paraxial rays DM≈AO+OC,D M \approx A O+O C, where AO≈h2/(−2s)A O \approx h^{2} /(-2 s) and OC≈h′2/2R.O C \approx h^{\prime 2} / 2 R . Besides, OB=OC−BC≈O B=O C-B C \approx ≈h′2/2R−h′2/2s′.\approx h^{\prime 2} / 2 R-h^{\prime 2} / 2 s^{\prime} . Substituting these expressions into (∗)(*) and taking into account that h′≈h,h^{\prime} \approx h, we obtain n′/s′−n/s=(n′−n)/Rn^{\prime} / s^{\prime}-n / s=\left(n^{\prime}-n\right) / R.
Prev
Next