The lens maker’s equation for a thin lens is easily derived from the equations for the focusing powers of the two surfaces as described in Equation (7.1). The situation is illustrated in Figure C.1, where the radius of curvature for the second surface, R₂, is a negative number.

Now imagine that the two diagrams are overlapped so that the lines defining the central plane of the thin lens coincide and the region having refractive index n₂ is confined between the two curves. The result is that the image plane of the first surface lies on the right side of the second surface at the position q₁. Therefore, the subject for the second optical surface can be defined as p₂ = –q₁. The equations for the powers of the surfaces are then given by:

FIGURE C.1 The two surfaces of a simple lens.

(C.1)

The equation for the thin lens is then obtained by adding these equations with n₃ = n₁. With p₁ = ∞, we have q₂ = f, thus we obtain the lens maker’s equation (Equation (7.3)):

(C.2)

Note that this equation is accurate only in the paraxial limit where the light rays make small angles with the optical axis and the lens is thin.

It is also interesting to consider the case where n₁ ≠ n₃. This time we find that:

(C.3)

Setting p₁ = ∞ establishes that the left-hand side of Equation (C.3) is equal to n₃/f₃, where f₃ is defined as the focal length in the region where the refractive index is n₃. Therefore, we obtain a new conjugate equation:

(C.4)

This equation, for example, applies to the human eye where the volume between the lens and the retina contains vitreous humor and n₃ = 1.337. By reversing the direction of the rays and setting q₂ = ∞, it is easy to show that n₁/f₁ = n₃/f₃.