# Lens Magnification

A modern photographic lens is the result of a complex design process at which one is aiming to maintain a high resolution, contrast, and possibly a pleasing bokeh, while keeping a number of unwanted effects at a minimum, such as geometric distortions, vignetting, chromatic aberrations, and flare. Other practical constraints such as cost, size and weight also have to be taken into account. This task is even more difficult for zoom lenses which have to provide a good balance over a (possibly wide) range of focal lengths.

To achieve these goals, a typical photographic lens is the combination of various optical elements (individual glass or plastic lenses) which correct for all the unwanted effects. Unfortunately, we usually do not know much about this internal design.

## Basic assumption

However, a complete photographic lens still behaves similar to a single thin lens which collects incoming parallel light rays into one single point. Thus, we make the simplifying basic assumption that we can model a photographic lens as a single thin lens. We shall later see how accurate this assumption really is.

## Focusing

A thin lens collects incoming parallel light rays into a focal point F. Its behaviour can be described by the well-known lens equation

 1 / f = 1 / g + 1 / h (M1)

where f is the focal length of the lens, g is the object distance between object and lens, and h is the image distance between lens and sensor (fig. 2). Only if this equation is satisfied, the picture of a point is again a point, and the image appears fully sharp.

When the lens is focused, it is actually moved back or forth a little such that the lens equation is satisfied, i.e. the distances g and h are adjusted. However, the focusing distance d which you can read and set on your lens measures the full distance between object and image plane, i.e.

 d = g + h (M2)

## Magnification

The magnification m in which we are interested here is defined as the relation between image size H and real object size G

 m = H / G (M3)

By similar triangles along the central ray (yellow), this is equivalent to

 m = h / g (M4)

To calculate m, we solve equation (M2) for h and substitute it into equation (M1)

 1 / f = 1 / g + 1 / (d - g) (M5)

With some algebra, we end up with a quadratic equation

 g² - g d + f d = 0 (M6)

This equation has (at most) two different real solutions, given by

 g = d / 2 ± sqrt (d² / 4 - f d) (M7)

where sqrt denotes the square root. With equation (M2), we also get

 h = d / 2 ± sqrt (d² / 4 - f d) (M8)

For simplicity, we define the root term as

 r = sqrt (d² / 4 - f d) (M9)

If we add r in equation (M7), we must subtract it in equation (M8), and vice versa. In practice, the vast majority of lenses are constructed such that the distance g between object and lens can be (much) larger than the distance h between lens and sensor. Otherwise, you could not focus at infinity (I am only aware of one such lens, the dedicated macro lens Canon MP-E 65mm). Thus, we set

 g = d / 2 + r (M10) h = d / 2 - r (M11)

With equation (M4), we can now easily calculate the magnification m of a lens as

 m = (d / 2 - r) / (d / 2 + r) (M12)

This formula is used for the lens magnification calculator. It is interesting to note that the magnification only depends on focal length and focusing distance of the lens, but not on other factors such as aperture or the size of the sensor. Results are thus valid for full frame, crop cameras and even mobile phone cameras alike. However, for a given magnification, you can of course capture a larger object with a larger sensor.

## Minimum focusing distance

Note that equation (M12) can only be solved if the expression under the root in equation (M9) is non-negative, i.e.

 d² / 4 - f d ≥ 0 (M13)

Division by d (positive for all meaningful cases) gives

 d ≥ 4 f (M14)

Since we are interested in the maximum magnification, it is clear that we must get as close to the object as possible. Equation (M14) defines a lower bound for the focusing distance d, i.e. we cannot get any closer than 4 times the focal length. How close we can actually get also depends on the mechanical construction of the lens. The closest possible distance dmin that still gives a sharp image is called the minimum focusing distance (MFD) of the lens. It is usually the smallest figure printed on the focus ring of your lens.

## Approximation

If we solve the lens equation (M1) for h and substitute it into equation (M4), we get

 m = f / (g - f) (M15)

This formula looks more simple than equation (M12) and is thus sometimes published to calculate the lens magnification, but remember that we usually do not know the object distance g, but only the focusing distance d. For close-up situations, these may differ significantly.

However, if the object is reasonably far away, i.e. d is much bigger than f, we can approximate (g - f) by g and g by d and get

 m ≈ f / d (M16)

Easy to remember, but for macro photography, the resulting figures are much too small.

## Notation

The magnification as derived by one of the given formulas is a positive real number. For example, for a focal length of 50 mm and a focusing distance of 0.3 m, we get a magnification of 0.27. This means that the size of the image H is 0.27 times (or 0.27×, 27%) of the size of the original object G.

In photography, a commonly used notation for lens magnification is

 1 : M (M17)

where M is calculated as the inverse of m. In our example, the magnification is 1 : 3.7, which indicates that the size of the image is 1 / 3.7 or about 1 / 4 of the size of the original.

May 10, 2014