This first appeared on my tech blog. I like to play around with various configurations of camera lenses. This partly is because I prefer to save money by using existing lenses where possible, and partly because I have a neurological condition (no doubt with some fancy name in the DSM-IV) that compels me to try to figure things out. I spent 5 years at an institute because of this problem and eventually got dumped on the street with nothing but a PhD in my pocket. So let this be a warning: keep your problem secret and don’t seek help.

A typical DSLR (or SLR) owner has a variety of lenses. Stacking these in various ways can achieve interesting effects, simulate expensive lenses (which may internally be similar to such a stack), or obtain very high magnifications. Using 3 or 4 lenses, a telextender, a closeup lens, and maybe some extension rings (along with whatever inexpensive adapter rings are needed), a wide variety of combinations can be constructed. In another entry, I’ll offer a companion piece of freeware that enumerates the possible configurations and computes their optical properties.

In the present piece, I examine the theory behind the determination of those properties for any particular setup. Given a set of components (possibly reversed) and some readily available information about them and the camera, we deduce appropriate optical matrices, construct an effective matrix for the system, and extract the overall optical properties – such as focal length, nearest object distance, and maximum magnification. We account for focal play and zoom ranges as needed.

The exposition is self-contained, although this is not a course on optics and I simply list basic results. Rather, I focus on the application of matrix optics to real camera lenses. I also include a detailed example of a calculation.

As far as I am aware, this is the only treatment of its kind. Many articles discuss matrix methods or the practical aspects of reversing lenses for macro photography. However, I have yet to come across a discussion of how to deduce the matrix for a camera lens and vice-versa.

After reading the piece, you may wonder whether it is worth the effort to perform such a calculation. Wouldn’t it be easier to simply *try* the configurations? To modify the common adage, a month on the computer can often save an hour in the lab. The short answer is yes and no. No I’m not an economist, why do you ask?

If you have a specific configuration in mind, then trying it is easier. However, if you have a set of components and want to determine which of the hundreds of possible configurations are candidates for a given use (just because the calculation works, doesn’t mean the optical quality is decent), or which additional components one could buy to make best use of each dollar, or which adapter rings are needed, or what end of the focal ranges to use, then the calculation is helpful. Do I recommend doing it by hand? No. I even used a perl script to generate the results for the example. As mentioned, a freeware program to accomplish this task in a more robust manner will be forthcoming. Think of the present piece as the technical manual for it.