Designing an Achromatic Doublet

The goal here is to develop a quantitative approach in designing an achromatic doublet that reduces chromatic aberrations.  We will have two lenses cemented together, the choice of lenses is up to you but lets keep it general.  The refractive power of each lens, by the lens-maker equation is..

We relate the lens to the Fraunhofer lines. The Fraunhofer lines are the dark lines in the emission spectrum of the sun, or the colored lines in the absorption spectrum. The refractive index of a material is actually not constant, but is a function of wavelength.  When we say the refractive index of glass for example, is 1.50 this is an average value of the index over the visible spectrum.  We use the Fraunhofer lines as a industry standard.  Light is shined through an element and certain absorption spectrums are produced.  When we are finished with the mathematical model, we test it at the broad ends of the spectrum and calculate the chromatic aberrations to see how good of a lens it is.

First thing we need to do is find the sum of the refractive power of the combined lens doublet.  This is simply the sum of the individual refractive powers if there is no space between them. (Since this is a doublet, there is no space between the lenses.)  Combining this idea with the former expressions...

We want the refractive power to be independent of wavelength, therefore the derivative of P with respect to wavelength should be equal to zero.

The partial derivative of the refractive index can be approximated using the F and C line of the Fraunhofer wavelength for the given material.

We now essentially want to expand the above expression to include the dispersion constants for the glasses used.  We then define a constant "V" which is defined as the reciprocal of the dispersive power.

From this, we can use the properties of the lens to find all four values of the radii!