# Depth Of Field

Definition: the distance between the nearest and furthest objects that can be imaged with reasonably sharp focus for a given focus setting

German: Schärfentiefe, Tiefenschärfe

Categories: general optics, vision, displays and imaging

Author: Dr. Rüdiger Paschotta

Most imaging instruments can provide sharp images only in a limited range of observation distances. Perfect imaging to a certain image plane (e.g., the location of an image sensor in a photo camera) is only possible for an object plane which is conjugate to the image plane. Image of objects before or after that conjugate object plane will be blurred to some extent. (An exception is the rarely used *camera obscura*.)

The width of the range of observation distances is called the *depth of field*. It must not be confused with the depth of focus, which is the corresponding quantity on the image side.

## Criteria for Reasonable Focus

For a quantitative definition of the depth of field, one requires a criterion for what level of defocus is acceptable. Different criteria may be sensible, depending on the situation:

- One may define the limits as the points where the imaging quality, measured for example with a point spread function, become significantly worse than in the focus. Such a criterion, however, may not be sensible if the imaging quality in the focus is much better than required or usable. For example, for an image sensor of a digital camera it would not matter if the circle of confusion rises from one tenth to one third of the pixel spacing.
- Therefore, one may alternatively define a certain diameter of the circle of confusion as the limit, irrespective of how small that circle can be for optimal focusing.

## Calculation of the Field of Depth

Here we use the second kind of defocus criterion as explained above, with a maximum diameter <$C$> of the circle of confusion, as calculated from purely geometrical optics. One then obtains the following equations for the nearest and furthest observation distance, when the focus is set to the distance <$d_\textrm{f}$>:

$${d_{{\rm{near}}}} = \frac{{{d_{\rm{f}}}}}{{1 + \frac{{\left( {{d_{\rm{f}}} - f} \right)\;C}}{{f\;D}}}}$$ $${d_{{\rm{far}}}} = \frac{{{d_{\rm{f}}}}}{{1 - \frac{{\left( {{d_{\rm{f}}} - f} \right)\;C}}{{f\;D}}}} \: \: {\rm{if }}\left( {{d_{\rm{f}}} - f} \right)\;C < f\;D \rm{, otherwise } \: \: \infty $$where <$D$> is the diameter of the aperture stop. It is assumed that we have a thin lens, where the entrance and exit pupil coincide with the lens. The depth of field is the difference between maximum and minimum distance.

Note that the above equations hold only for small angles; that condition is usually reasonably well fulfilled in photography, for example, but not for microscopes.

When focusing to short distances, the depth of field will be relatively small. When focusing to larger distances, it increases, and one eventually reaches the hyperfocal distance where the maximum distance becomes infinity. That is the situation where the depth of focus is the largest possible.

As is well known in photography, the depth of field can be increased by reducing the diameter of the aperture stop, which at the same time decreases the image brightness (or requires a correspondingly longer exposure time).

The presented equations hold only within the validity of geometrical optics, which is however usually given in the context of photography.

See also: imaging, imaging with a lens, hyperfocal distance, depth of focus

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