A modern camera lens is the result of a complex design process that focuses on high resolution, high contrast, a large maximum aperture, a pleasant bokeh and possibly a high magnification, while minimizing a number of undesirable effects such as geometric distortions (straight lines appear curved), chromatic aberrations (color fringes at high contrast boundaries), vignetting (darkened image boundaries) and flare (reflections within the optical system). Other practical constraints such as size, weight and cost must also be taken into account. 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 the 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 provided by the manufacturers such as
- 5 elements in 4 groups (for the HD Pentax DA 40 mm f/2.8 Limited pancake lens)
- 23 elements in 18 groups with 1 XA, 2 asperical, 2 Super ED and 5 ED elements (for the Sony FE 70-200 mm f/2.8 GM OSS professional grade zoom lens)
which give us an idea of the complexity of the construction, and possibly a lens diagram illustrating the shape and position of the elements, we do not know much about the optical design of a lens.
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 are a number of variables beyond our control or knowledge. Furthermore, this requires a little math. But fear not, most of the equations are about similar triangles and basic algebra, so it hardly ever exceeds the mathematics 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 approximations that are much easier to understand and remember.
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 (be extremely careful if you do any experiments with sunlight!!!!!). 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.