# Geometric optics multiple lens focal length calculation

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!

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.

• Thanks, will have a crack at writing out an equation for EFL of the 8 lenses. I'm still a bit confused with how to determine the focal length of the individual lenses in this setup though, any advice for how to tackle that? Commented Feb 10 at 9:06
• Also which of those lenses would be considered thick lenses, is it just the roundish one (6th one) Commented Feb 10 at 11:23
• @user94863 It's not clear to me why you are doing this calculation. You physically have the lenses available, you are planning to buy them or it's all just on paper? I ask this because if you buy lens for a manufacturer, they will provide you with all the information; if you physically have the lenses in your hands there are some really quick tricks to calculate $f$. On the other hand, if you are only doing it on paper, I don't know any other way but the equation. Commented Feb 10 at 18:49
• For "thin" lenses usually physicist consider those that have thickness "negligible" compared to their size. I agree with you that it is a really sketchy definition and may be difficult to apply. The fact is that no lens is "thin" but the key factor is to consider what happens to the ratio $\frac{d}{f_1f_2}$. If $d$ is 1% of $f_1f_2$ you will get your answer wrong only by more or less that percentage. It really all comes down to how much you need your calculations to be precise Commented Feb 10 at 18:55
• I don't have the lenses, I just saw this figure and wanted to simply understand how they managed to calculate an effective focal length of 5mm with that lens arrangement. Most if not all effective focal length calculations I've seen are of much simpler arrangements than this, just wanted to understand the maths behind it Commented Feb 10 at 20:10

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).

• I wish I knew this technique for my optic lab exam... the professor had us to compute focal length of some lens system lens by lens using my method. As you say, it's really easy to make a mess. Commented Feb 10 at 18:59
• Will check this out too, thanks Commented Feb 10 at 20:18