Geometric optics multiple lens focal length calculation

Question

Just wanted to understand how I would go about calculating the effective focal length of these different lens configurations.

There's the combined focal length formula, but I'm struggling to apply it to this setup, and how would I account for the different lens types other than the bog standard concave and convex ones in simple examples that also only consider two lenses, rather than the eight here.

Thanks!

Answer 1

To describe the behaviour of lenses (assuming their "round part" is spherical") is is possible to use the Lens-maker' equation.

This formula takes into account the most important factors in the behaviour of a lens: in particular the radii of the surfaces (concave or convex), its thickness and the material used to make the lens.

So to answer to the second part of your question

how would I account for the different lens types other than the bog standard concave and convex ones

The answer is that to calculate what a lens that is not "standard" does you have to know these parameter and apply the formula. You can find it easily in Wikipedia: https://en.wikipedia.org/wiki/Lens#Lensmaker's_equation

For what regards the first part, calculate the combined focal length is a quite tedious process. If the lenses are thin you can more or less apply the following equation. $$ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1f_2} $$ Where the $f$s are the focal length of each lens and $d$ their distance.

Is is a little more tricky when you have more chunky and thick lenses. In this case you can calculate the focal length using the Lens-maker' formula and sum but you will need to chose $d$ as the "effective distance" between the lenses as shown in the image.

The following video may help you: https://www.youtube.com/watch?app=desktop&v=vQFN8EzrHqE

Answer 2

One can theoretically reduce a pair of lenses to a single lens. By repeatedly applying this process, you could reduce your whole lens system to a single lens, and extract the focal length. However, this is non-trivial - it gets messy because one has to keep track of the principal planes, which are theoretical planes in a lens element where the rays refract. Unless you are familiar with principal planes, you are likely to make a mistake.

A better way to do this is to construct what is called a paraxial ray trace. Paraxial refers to rays that are infinitesmally close to the optical axis. The basic method is to send in a ray from infinity, and then trace it, using paraxial equations for refraction (at each surface) and transfer (between surfaces). The refraction and transfer equations are simple and intuitive. You will need to know the lens curvatures, glass type (i.e. refractive index) and thicknesses. The advantage of the paraxial ray trace it is easier to see if you make a mistake. A good explanation of this process is here:

https://www.edmundoptics.com/knowledge-center/application-notes/optics/geometrical-optics-101-paraxial-ray-tracing-calculations/

(See about 1/2 way down the page).