## Lens and Depth of Field

Lenses are characterised by their focal length and the maximum size of the aperture, measured as an “f/number”.

### Focal length

The focal length of a lens is the distance between the centre of the lens and the plane of the camera’s sensor when the lens is focussed on a distant object.

Wide angle | Normal | Telephoto |
---|---|---|

28mm or less | 35 to 50mm | 85mm or more |

Landscapes | General purpose | Portraits, nature |

The focal length determines the angle of view of a lens. For a standard SLR camera, a lens with focal length of about 35mm to 50mm gives a view that seems most “natural”. Lenses with shorter focal lengths give a wide field of view, “wide-angle”, and objects in the middle distance appear to be smaller / further away. In the other direction, long “telephoto” lenses have a narrow field of view and objects appear closer – like looking through binoculars.

### f/number

The f/number expresses the lens aperture as a fraction of its focal length. For example if a lens is described as “50mm f/2”, then the maximum aperture is 50/2, i.e. 25mm.

Most lenses have a variable aperture and, as the lens is “stopped down”, the f/number continues to describe the size of the aperture as a fraction of its focal length. For example, if the width of the aperture is set to an eighth of the focal length, the aperture is called “f/8”. Notice that as the size of the aperture is reduced, the f/number gets bigger; this is a little counter-intuitive until one gets used to it.

### Depth of field

When you focus the camera on any subject, there is always a band, stretching from in front of the subject to behind, in which all objects are in acceptably-sharp focus. This band of good focus is the “depth of field”. It is affected by lens aperture, focal length and distance to the subject as in this table:

Shallow depth of field | Deep depth of field |
---|---|

Telephoto lens | Wide-angle lens |

Subject close to camera | Subject in the distance |

Wide aperture (low f/number) | Small aperture (high f/number) |

The pictures below show the effect of changing the f/number. The scene was taken with a standard 50mm lens at f/1.2 (first photo) and then f/16 (second photo), both focussed on the sign.

### Hyperfocal distance

An interesting if somewhat technical concept is that of hyperfocal distance. The idea here is to imagine taking a photo of a landscape and focussing on the distant horizon. In this scenario, the band of acceptable focus will stretch from the middle distance to a point well passed the horizon. Our depth of field will extend beyond the farthest limits of the scene. So we can conceive of a distance at which we might focus the lens so that the far end of the depth of field just extends to the horizon. This is the hyperfocal distance.

For example, if I take a 50mm lens at f/8 and focus on the far horizon, the depth of field will extend from 9.75 metres in front of me to infinity (and beyond) – Buzz Lightyear would be proud. However, if I now focus at 9.9 metres, the depth of field will extend from 4.9 metres to (just) infinity (see footnote). So I get much more of the foreground in focus while still keeping the far end of the depth of field at infinity. It is clearly inconvenient to have to calculate such things but we can make use of the concept by focussing in the middle distance and using the camera’s depth of field preview button to check that the depth of field extends far enough into the distance.

Note that this concept should be used with care. When talking about depth of field, we are referring to the area of acceptable focus, not critical focus. In the example above, if it is important to get the horizon as sharp as possibe, we should focus on the horizon.

### More on depth of field

A demonstration tool for calculating depth of field is provided: Depth of field calculator

When the subject distance is much less than the hyperfocal distance and the magnification is small (i.e. not macro), an approximation to depth of field is given by:

Where N is f/number, c is acceptable circle of confusion, f is focal length and s is distance to subject.

For details of the mathematics, see: Depth of field derivation

**Effect of f/number.** From the expression above, we can see that the effect of f/number (N) is straightforward. DOF is proportional to the f/number set on the lens – doubling the f/number doubles the DOF.

**Effect of Focal Length.** The expression above indicates that DOF is inversely proportional to the square of the focal length, i.e. doubling focal length reduces the DOF by a factor of four.

But what happens if we adjust both focal length and distance to subject? If we take a given camera (thus defining c) and set up a head and shoulders shot, we can maintain the same magnification across a variety of focal lengths simply by changing the distance such that (s/f) remains constant. The DOF then depends only on f/number (N). This gives rise to the common observation that for a fixed composition, the DOF is independent of the focal length. While this is something of a half-truth, it is a point worth noting.

**Effect of Sensor Size.** To understand the effect of sensor size, we have to consider the **“acceptable circle of confusion”** (c) in the expressions above. Consider photographing a single, very small point of light. If we focus exactly on that point, the image created on the sensor will also be a tiny point. If we then focus slightly in front or behind the point of light, the image on the sensor will be a small, blurred circle. The question is, at what amount of out-of-focus will we start to notice the blurring..? The “acceptable circle of confusion” is defined as the largest blurred circle on the sensor that, when viewed in normal conditions in the final print, will still be perceived as a point. Normal conditions is sometimes defined as viewing an 8″ by 10″ print at 25 centimetres, i.e. hand held.

If we start with the (physically small) image on the sensor and create an 8″ by 10″ print, the amount of magnification involved depends on the relative size of the sensor and the print. If the sensor were 1″ wide, the image would be magnified 10 times to fill a 10″ print. If our small point of light was slightly out-of-focus and was recorded as a small blurred circle on the sensor, the blurred circle will be 10 times bigger on the print. But will we actually see it as a point or as a blurred circle?

Starting with the print, let's assume that the smallest circle we can identify as a circle is about 1/125″ - anything smaller will be seen simply as a dot. With a 10-times magnification between sensor and print, this circle would be just 1/1250″ on the 1″ sensor. So the acceptable circle of confusion for the 1″ sensor would be 1/1250″ or 0.02 mm.

Now switch to a camera with a 2″ sensor. The magnification between sensor and print is now just 5-times - and a circle of 1/125″ on the print will correspond to a circle of 1/5th of that size on the sensor - i.e. 1/625″ or 0.04 mm. So the acceptable circle of confusion for the 2″ sensor is twice that for the 1″ sensor.

The acceptable circle of confusion is therefore proportional to the size of the sensor. Examples of acceptable circle of confusion for common sensors sizes are:

- ♣ 0.029mm for 35mm sensor
- ♣ 0.018mm for APS-C Canon
- ♣ 0.015mm for Four Thirds

Note from the expressions above for DOF that if we set up a studio shot with a given subject distance (s), focal length (f) and f/number (N), the DOF is proportional to the acceptable circle of confusion (c) and therefore proportional to the sensor size. A larger sensor gives a larger DOF.

However, consider setting up a full-frame 35mm camera and a Four-Thirds camera side by side, with the same f/number, and choosing lenses to get the same field of view, perhaps a head-and-shoulders shot. If we choose a standard 50mm lens on the full frame camera, we would need a 25mm lens on the Four-Thirds camera to reproduce the same field of view on its smaller sensor. The effect of the smaller sensor on the Four-Thirds camera would be to halve the DOF. But the focal length used on the Four-Thirds camera has also been halved - and halving the focal length should increase the DOF by a factor of four. The effect of focal length wins and the overall effect is a doubling of the DOF for the Four-Thirds camera.

It is frequently said that a characteristic of cameras with small sensors is that they have large DOF. This again is a half-truth. The direct effect of reducing the size of sensor is to reduce the DOF. However, as described above, cameras with small sensors are generally used with lenses of short(er) focal lengths which, for a given composition, increases the DOF.

For a discussion of equivalence of different sensor formats, see: Equivalence

**Footnote** - hyperfocal distance. When the lens is focussed at the hyperfocal distance, the near focus distance will be half the hyperfocal distance. This follows from the mathematical explanation at: Depth of field derivation