Editing Pinhole camera (section)

===Basis for optimum pinhole size===
[[File:RL of PH Camera vs FL.jpg|thumb|upright=2|A graph of the resolution limit of the pinhole camera as a function of focal length (image distance)]]
In the 1970s, Young measured the resolution limit of the pinhole camera as a function of pinhole diameter<ref name=Young-71>{{cite journal |last1=Young |first1=M. |title=Pinhole Optics |journal=Applied Optics |date=1 December 1971 |volume=10 |issue=12 |pages=2763–2767 |doi=10.1364/ao.10.002763 |pmid=20111427 |bibcode=1971ApOpt..10.2763Y }}</ref><ref>{{cite journal |last=Young |first=M. |title=Pinhole Imagery |date=May 1, 1972 |volume=40 |issue=5 |pages=715–720 |journal=American Journal of Physics |doi=10.1119/1.1986624}}</ref> and later published a tutorial in ''The Physics Teacher''.<ref>{{cite journal |last1=Young |first1=Matt |title=The pinhole camera: Imaging without lenses or mirrors |journal=The Physics Teacher |date=December 1989 |volume=27 |issue=9 |pages=648–655 |doi=10.1119/1.2342908 |bibcode=1989PhTea..27..648Y }}</ref> He defined and plotted two normalized variables: the normalized resolution limit, <math>\frac{RL}{s}</math>, and the normalized focal length, <math>\dfrac{f}{\left ( \frac{s^2}{\lambda} \right )}</math>, where
: {{mvar|RL}} is the resolution limit
: {{mvar|s}} is the pinhole radius ({{mvar|d}}/2)
: {{mvar|f}} is the focal length
: {{mvar|λ}} is the wavelength of the light, typically about 550&nbsp;nm.

On the left-side of the graph (where the normalized focal length is less than 0.65), the pinhole is large, and geometric optics applies; the normalized resolution limit is approximately constant at a value of 1.5, meaning the actual resolution limit is approximately 1.5 times the radius of the pinhole, independent of the normalized focal length. (Spurious resolution is also seen in the geometric-optics limit.)

On the right-side (normalized focal length is greater than 1), the pinhole is small, and [[Fraunhofer diffraction]] applies; the resolution limit is given by the far-field diffraction formula shown in the graph, which increases as the pinhole size decreases, assuming that {{mvar|f}} and {{mvar|λ}} are constant:
:<math>RL = \frac{0.61 \cdot \lambda f}{s}</math>
In this version of formula as published by Young, the radius of the pinhole is used rather than its diameter, so the constant is 0.61 instead of the more usual 1.22.

In the center of the plot (normalized focal length is between 0.65 and 1), which is the region of near-field diffraction (or [[Fresnel diffraction]]), the pinhole focuses the light slightly, and the resolution limit is minimized when the normalized focal length is equal to one. That is, the actual focal length {{mvar|f}} (the distance between the pinhole and the film plane) is equal to <math>\frac{s^2}{\lambda}</math>. At this focal length, the pinhole focuses the light slightly, and the normalized resolution limit is approximately {{frac|2|3}}, i.e., the resolution limit is ~{{frac|2|3}} of the radius of the pinhole. The pinhole, in this case, is equivalent to a Fresnel zone plate with a single zone. The value {{mvar|s}}<sup>2</sup>/{{mvar|λ}} is in a sense the natural focal length of the pinhole.{{fact|date=April 2024}}

The relation {{mvar|f}} = {{mvar|s}}<sup>2</sup>/{{mvar|λ}} yields an optimum pinhole diameter {{mvar|d}} = 2{{radic|{{mvar|fλ}}}}, so the experimental value differs slightly from the estimate of Petzval, above.