Depth of field (DOF).
DOF is the range of subject distances for which an image is acceptably well focused. What do we mean by "acceptable" focus? We shall define this more precisely later, but for now, let's just say that focus is acceptable if the human eye cannot detect any fuzziness in the final print, at least in the areas that are intended to be sharp.
Look at the figure below, which shows light rays from a subject at the right passing through a theoretical 'thin' lens (at O, the zero point) to the image plane (film or digital sensor) at the left.
In this diagram, the (uppercase) D's are subject distances from the lens, while the subscripts mean:
0 = Exact focus point,
1 = Near limit of acceptable subject focus,
2 = Far limit of acceptable subject focus.
The (lowercase) d's are image distances from the lens, with the subscripts (0, 1, 2) corresponding to the D subscripts already defined. Both the subject distances (the D's) and the image distances (the d's) are measured from the optical centre of the lens at 'O'.
We have drawn a theoretical "thin" lens to simplify matters. Real camera lenses are not like this: they have multiple elements and significant physical and optical 'thickness'. Their detailed analysis requires use of two optical "centres", called the front and rear nodal points, separated by the optical thickness of the lens, but this makes no difference to the analysis of DOF.
We shall also use the following additional symbols, not shown in the figure above:
f = Focal length of the lens;
a = Aperture diameter of the lens diaphragm;
c = Circle of confusion diameter (discussed below);
x = "f/stop" setting of the lens.
Note that x = f/a by definition, so a = f/x.
To summarize, this is a simplified diagram of a lens with some rays of light passing through it from a point subject at the right, to the film (or sensor) plane at the left. The optical centre of the lens is at the zero point: the vertical line marked 'O'. The exactly focused point of the subject is at a distance D0 to the right of the lens, and all the rays from this subject pass through our (theoretically perfect) lens and meet on the film plane at a distance d0 to the left of the lens. For this analysis we shall disregard complications such as the possibility of optical flaws or aberrations in the lens.
Now the image of a point object at D_{1} (nearer the camera than D_{0}) will fall behind the film plane at
d_{1} (to the left of d_{0} in the diagram). The light rays that form this image will fall within a circle where they pass through the film plane, and the diameter of this circle is the distance between the top and bottom rays passing through the film plane on their way to the point
d_{1}. The image of the point D_{1}
will therefore be blurred, but it will be sufficiently in focus to be perceived as sharp if the diameter of this circle, called the circle of confusion, is less than or equal to some tiny distance
c. How small this distance c needs to be depends on the degree of enlargement in the final print and on the sharpness of human eyesight, as we shall see.
Similarly, the image of a point object at D_{2} (further from the camera than
D_{2}) will fall in front of the film plane at
d_{2} (to the right of it in the diagram). The light rays that form this image will continue on to the film plane, however, and will be blurred into a circle where they reach it. Once again, this image will be sufficiently in focus to be perceived as sharp if the diameter of the circle of confusion is less than or equal to
c. So if we set the range of subject distances from D_{1}
to D_{2} so that in both cases the circle of confusion at
d_{0} has diameter c, then D_{2}
 D_{1} is our depth of field. All the subject matter between these points is "acceptably" in focus. We shall come back to the question of how to determine the size of the circle of confusion and what we mean by acceptable focus later.
The rays we have drawn in the diagram are those that pass through the edges of the lens. The effective diameter of the lens is
a. The vertical 'spread' of the rays passing through the film plane (on the line at
d0) is the diameter of the circle of confusion c. From the geometry of the triangles to the left of the lens, we can now establish the following relationships:


(1)* 

(2)* 
The derivation of all the (*) starred equations will be found at the end of this article.
We now recall the fundamental lens formula from our highschool physics classes:


(3) 
Here, i represents any single one of the subscripts (0, 1 or 2) used in the
diagram.
We can now establish relationships between
D_{0}, D_{1} and D_{2}. Firstly:


(4)* 
When D_{2} is very large ("approaches infinity"), the
af term in the denominator becomes insignificant in relation to the
cD_{2} term, so D_{0} approaches
This limiting value of D_{0}
is known as the hyperfocal distance, which we shall denote by the symbol
h. This is the nearest exact focus distance at which the infinity point is just in acceptable focus. Any higher value of
D_{0} would result in the value of D_{2}
being undefined ("greater than infinity").


(5a) 
Setting the focus point at the hyperfocal distance gives the greatest possible DOF: it makes the near point of acceptable focus as close as it can be, while the far point of acceptable focus remains at infinity.
In normal photographic situations, a
is significantly larger than c, and a = f/x, where
x is the "fstop" setting of the lens, so we may write formula (5a) approximately as:


(5b) 
This tells us that the hyperfocal distance is roughly proportional to the square of focal length, and inversely proportional to the aperture number and to the diameter of the circle of confusion. Formula (5b) is the one given by Ansel Adams in "The Camera" (appendix).
We now require expressions for D_{1} and D_{2} in terms of
D_{0}. The first of these gives:


(6a)* 
For normal photography, c is much smaller than
a, and f is much smaller than D_{0}, so this result reduces to:


(6b) 
The formula for D_{2} comes immediately from rearranging formula (4) above:


(7a)* 
The approximate result corresponding to this is clearly:


(7b) 
Formulas (6b) and (7b) are also equivalent to those given by Ansel Adams (op.cit.).
Let us now estimate a reasonable value for c. Assume we are using 35mm film with an image size of 24 x 36mm. Let us further assume that we wish to enlarge this negative to a print size of 8 x 10 inches, which is well within the potential of 35mm film. This implies a magnification of about 8x from film to print.
Now we know (see the Resolution article) that the resolution of the human eye can reach about 10 line pairs per mm (equivalent to 20 pixels per mm in digital terms, or about 500 pixels per inch) under ideal lighting conditions when scrutinizing a print from a distance of about 7 inches. But the normal viewing distance for a print this size would be about 13"  the length of the print diagonal  so we may reasonably accept a print resolution of about 5 lp/mm. With 8x enlargement, we therefore require a resolution of about 40 lp/mm on the film. On this basis, the required upper limit for
c would be 0.025mm. In actual practice, the standard value of c that is normally used for depth of field tables for 35mm format ranges from 0.03 to 0.035mm. For the following discussion we shall use a value of 0.03mm for
c.
Let us also assume for now that we have a moderate telephoto lens with a focal length
f of 100mm (0.1 metres), and assume a typical midrange aperture setting of
x = 5.6. Then formula (5b) above reduces to:


metres 
When D_{0} = h, from (5a) and (6) we can show that:


(8)* 
This tells us that when focus is set at the hyperfocal distance, so that the far point of acceptable focus is at infinity, then the near point of acceptable focus is half the hyperfocal distance. This is a very important result that has often been misstated and misunderstood.
Thus, under the assumptions used previously, a 100mm lens can be set at an aperture of f/5.6 to have depth of field from 59.5 / 2 = 29.75 metres out to infinity, with the point of exact focus at the hyperfocal distance: 59.5 metres. Clearly, the DOF behind the point of exact focus is infinite. The DOF in front of the point of exact focus is 29.75 metres.
With focus set at this distance, giving the maximum possible total DOF, the ratio


(9) 
is infinite, because D_{2}
is infinite.
However, when the focus point is closer than the hyperfocal distance, the DOF beyond the point of focus is no longer infinite. For any value of
D_{0} less than the hyperfocal distance h, we can calculate how the ratio of these distances, given in equation (9), varies.
The general formula is obtained from equations (6) and (7):


(10)* 
With the same assumptions used previously, but setting the exact focus point
D_{0} to 30 metres, as an example, our (approximate) formulas give:


metres 


metres 
and the depth of field is D_{2}
 D_{1} = 40.6 metres.
As we reduce D_{0}, the ratio falls, until when D_{0} =
f, then
When D_{0} < f, the ratio is less than one.
We sometimes hear a rule of thumb claiming there is twice as much depth of field behind the point of focus as there is in front of it. This is clearly not accurate, but may be used as a very rough approximation for "normal" photographic distances (whatever that might mean).
When is the ratio equal to 2, as claimed by the rule of thumb? From (10), it is when
... which is when:


(11)* 
Using the same assumptions as for our previous examples, this gives:


metres, 
almost exactly onethird of the hyperfocal distance. There is no special meaning in this result, except that 20 metres may possibly be regarded as a typical midrange focusing distance for a 100mm lens (in the 35mm format), thus providing (rather tenuous) grounds for the rule of thumb.
As we have seen, depth of field is given by
for values of D_{0}
< h. For values of D_{0} much smaller than
h, this formula reduces to
In this range, depth of field increases roughly in proportion to the square of subject distance, but at greater ranges, DOF increases even faster than this, and becomes infinite, as already mentioned, for values of
D_{0} ≥ h.
We may also point out that DOF is inversely proportional to the hyperfocal distance, which means, in effect, that:
1. DOF is proportional to c, diameter of the circle of confusion;
2. DOF is inversely proportional to a, the actual aperture of the lens;
3. DOF is inversely proportional to f, focal length of the lens.
However, for practical purposes, the diameter c of the circle of confusion that we require is proportional to the diagonal of the image. Less enlargement is needed with larger formats to achieve the print dimensions required.
For any required angle of view, moreover, focal length is also proportional to the size of the image. Thus for any given composition (camera placement and angle of view), we have less DOF at any given aperture with larger formats.
Formula derivations
(1) 

(2) 

(4) 

(6a) 

(7a) 

(8) 

(10) 

(11) 

