Consider the case of a thin lens, created by grinding off the second side of a spherical surface. We begin by deriving the relationship between between the path of incident and twice refracted light from an initial medium of index of refraction n_l through a thin lens with radii of curvature R₁ and R₂ and index of refraction n_m.

[NMSU, N. Vogt]

For the first surface of refraction, we use the results of our analysis of refraction through a spherical surface.

The second surface "sees" the light transmitted through the lens as incoming light from the virtual image located at s_o2.

We can use the fact that |s_o2| = |s_i1| + d, and combine the two equations of refraction to obtain the following.

[NMSU, N. Vogt]

For the case of air, this is the lenmaker's formula.

For the case of an object at an effective distance of infinity, this simplifies to

[NMSU, N. Vogt]

For the case of an object imaged at infinity, this simplifies to

[NMSU, N. Vogt]

where we define the thin-lens formula