I am trying to follow the derivation of lens maker's formula from the textbook "University Physics", p.1133 (https://books.google.com.hk/books?id=nQZyAgAAQBAJ&pg=PA1133#v=onepage&q&f=false)
I can understand the first equation because it is just the object–image relationship for spherical refracting surface. But for the second equation, why the left hand side is nb/s2+nc/s'2 instead of nc/s2+nb/s'2? s2 is the first image's distance and it is on the nc side. In addition, on the right hand side why it is nc-nb on the numerator instead of nb-nc? If we follow strictly the formula for spherical refracting surface, the nb should be the lens side and nc is the air side.
A more fundamental question is, why this kind of superposition principle can be applied? I mean why the lens can be expressed as two lens added together? In many books they directly apply the object–image relationship for spherical refracting surface twice and added together. But this formula is only for single spherical surface (e.g. one side is air only and the other side is water only). If it is a lens it is air on both sides but lens in the middle. Why the solution for single spherical surface can be superposed like this?
Best Answer
The superposition is only approximately correct and the easiest way to understand both (1) why it works and (2) when it can be applied is to think in terms of waves and wavefronts, not rays. Since your link isn't working, let's write the equation down:
$$P_{lens}\approx \frac{n_{lens}-n_0}{n_o}\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$
With the fields thought of as waves, the lens surfaces become phase masks and the superposition of the two lens curvatures holds because the phase delays imparted by the phase masks one after the other are additive as long as the wavefront curvature (i.e. lateral phase distribution of the field) does not change much between the phase masks.
The power of the lens is the reciprocal of its focal length $f$, i.e. it is the reciprocal of the radius $f$ of curvature of the wavefronts that are output from the lens when a plane wave is input. A spherical wave of this radius converges to its diffraction limited focus after having propagated through this distance. So think of a plane wave input: the first surface represents a phase mask with phase delay as function of distance $r$ from the optical axis given by:
$$\frac{2\,\pi}{\lambda} (n_{lens}-n_0) \frac{r^2}{2 R_1}$$
the phase mask function for the second is:
$$-\frac{2\,\pi}{\lambda} (n_{lens}-n_0) \frac{r^2}{2 R_2}$$
and so the total phase delay is simply:
$$\frac{2\,\pi}{\lambda} (n_{lens}-n_0) \frac{r^2}{2} \left(\frac{1}{R_1}-\frac{1}{R_2}\right)$$
Now ask yourself: what is the radius of curvature of the spherical wave that shows this (to terms of second order and lower) phase distribution. You'll find it is the reciprocal of the power given by the lensmaker's formula.
I give the full details of these kinds of calculations in this answer here. I show the calculation above as well as the transformer matrix method where one thinks of lens surfaces and other optical processing elements as operators in the group $SL(2,\,\mathbb{R})$.