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2.8: Derivation of the Powers

Up to this point I have defined what is meant by convergence, and I have defined power as the difference between the final and initial convergences. I asserted without proof formulas for the powers of a lens, a refracting interface, and a mirror. It is now time to derive them. Remember that in this chapter I am dealing with small angles only (indeed if angles are not small, a point object will not result in a point image) and consequently I am going to assume that any angle is equal to its \tangent or to its \sine, and I am going to write Snell’s law in the form

\( n_1 \sin \theta_1 = n_2 \sin \theta_2\) or \(n_1 \tan \theta_1 = n_2 \tan \theta_2\)  or \(n_1 \theta_1 = n_2 \theta_2\)

as the spirit moves me and at my convenience.