Circle of confusion computation
Note: Only archiving this here before it is lost.
What follows is a computation of the Circle of confusion using "modern" notation and showing all of the steps.
<math>A = \mbox{aperture diameter}</math>
<math>f = \mbox{focal length}</math>
<math>N = \mbox{f-number} \equiv \frac{f}{A}</math>
<math>s = \mbox{distance from lens to point source}</math>
<math>s + \Delta s = \mbox{distance from lens to external focal plane of sensor}</math>
<math>l = \mbox{distance from lens to internal focal plane of point source}</math>
<math>l - \Delta l = \mbox{distance from lens to sensor plane}</math>
<math>B =\mbox{circle of confusion diameter at sensor}</math>
<math>B = \frac{A}{l} | \Delta l |</math> (trigonometry, neglecting diffraction)
<math>\frac{1}{l} + \frac {1}{s} = \frac{1}{f}</math> (basic lens equation)
<math>\frac{1}{l - \Delta l} + \frac {1}{s + \Delta s} = \frac{1}{f}</math>
<math>l = \frac{1}{\frac{1}{f} - \frac{1}{s}}</math>
<math>\Delta l = l - (l - \Delta l) = \frac{1}{\frac{1}{f} - \frac{1}{s}} - \frac{1}{\frac{1}{f} - \frac{1}{s + \Delta s}} = f \left ( \frac{1}{1 - \frac{f}{s}} - \frac{1}{1 - \frac{f}{s + \Delta s}} \right )</math>
<math>s \gg f, s + \Delta s \gg f</math> (far field approximation)
<math>\Delta l \approx f \left [1 + \frac{f}{s} - \left ( 1 + \frac{f}{s + \Delta s} \right ) \right ] = f^2 \left ( \frac{1}{s} - \frac{1}{s + \Delta s}\right ) = f^2 \frac{\Delta s}{s (s + \Delta s)}</math>
<math>l \approx f \Rightarrow \frac{A}{l} \approx \frac{1}{N}</math>
| \Delta s|}{s (s + \Delta s)}</math> |
To see what effect the sensor size has on the size of the circle of confusion, we need to rewrite this in terms of the effective circle of confusion and the effective focal length. For a given camera position and a given framing, the effective focal length is independent of the sensor size, as opposed of the actual focal length, which is proportional to the sensor size. The effective circle of confusion is the actual circle of confusion as a fraction of the sensor size, such that for a given print size and printed circle of confusion, the effective circle of confusion is independent of the sensor size.
<math>B_{eff} = \frac{f_{eff}}{f}B \approx \frac{f}{f_{eff}} \cdot \frac{f_{eff}^2}{N} \cdot \frac{|\Delta s|}{s (s + \Delta s)} = \frac{f_{eff}^2}{N_{eff}} \cdot \frac{|\Delta s|}{s (s + \Delta s)}</math>
<math>N_{eff} = \frac{f_{eff}}{f} N</math>
This confirms the assertion in the Depth of field entry that the depth of field is a function of the actual aperture size, independent of the sensor size.
