In this paper, deduction of the effective focal length for aA spherical thick lens inmersed in two diferent, of index of refraction na, immersed between two different optical media, with index of refraction n1 and n2, consist in two refractive spherical surfaces of radios R1a and R2a, separated a distance da over the optical axis, see Figure bellow:
In the figure, a generic optical ray (in brown color) is shownpropagating in the first media with index of refraction n1 with a alpha1 angle with respect to the optical axis. Then the ray is refracted by the first surface of the lens, at a height y1. After that the actual formula you asked forray propagates inside the lens with refraction index na (a distance da measured over the optical axis) and is refracted by the second surface, leaving the lens with an angle of alpha2 at a height y2 with respect to the optical axis, following further propagation into the n2 medium. The mathematical relation between angles and heights in the two surfaces can be obtained simply makingusing the optical matrix formalism in the following way:
Performing the matrix product:
The expression corresponding to the matrix element a12 is:
Which is the focal distance for the thick lens. Making n1=n2=1, it is obtained the expression for the focal distance (thickor effective focal distance, as is called by Hetch in "Optics"), for a thick lens inmersed in air.) See http://pubs.sciepub.com/ijp/9/6/1/ Also contains deduction for systems consisting of two and N thick lenses. I hope it can be usefullmore details.