# 1 Introduction

A modern camera lens is the result of a complex design process. A number of desirable characteristics such as a high resolution, high contrast, a large maximum aperture and a pleasing bokeh must be optimized. Equally important, various undesirable effects such as geometric distortions (straight lines appear curved), chromatic aberrations (color fringes at high contrast boundaries), vignetting (darkened image corners) and flare (reflections within the optical system) need to be minimized. Other practical constraints such as size, weight and last not least cost of the lens must also be considered. This task is even more difficult for a zoom lens, which has to provide a good balance over a possibly wide range of focal lengths.

To achieve these goals, a typical camera lens is a combination of several optical elements (individual glass or plastic lenses) which correct for all the unwanted effects. All these elements are hidden inside the lens barrel, so except for some rather general descriptions by the lens manufacturers such as

• 5 elements in 4 groups (smc Pentax DA 40 mm f/2.8 XS pancake lens)
• 27 elements in 19 groups (Sigma 60-600 mm f/4.5-6.3 DG DN OS | Sports zoom lens)

and possibly a lens diagram illustrating the approximate shapes and positions of the elements, we do not know much about its optical design.

## 1.1 What these pages are about

On these pages, we will derive the formulas used for the magnification and depth of field calculator. This seems a little questionable since we have just seen that there is a number of variables beyond our knowledge or control. Furthermore, it requires a little math. But fear not, most of the equations are about similar triangles and basic algebra, so they hardly go beyond the math curriculum of the ninth grade. You can decide for yourself if you believe what you see, or prefer to check some or all of the transformations. Wherever possible, we will also derive more intuitive approximations.

## 1.2 Basic assumption

As can easily be verified, a camera lens still behaves like a single convex lens, as it collects incoming parallel light rays in one single focal point. Therefore, we make the simplifying basic assumption that a camera lens can be modeled as a single thin lens, which is described by the well-known lens equation. We shall later see how accurate this assumption really is.