Why do pinhole-size formulas use different constants?
Asked 1/3/2014
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I’m trying to understand the standard pinhole diameter formula:
d = c√(fλ)
where d is the pinhole diameter, f is the distance from the pinhole to the film/sensor, and λ is the wavelength of light. Using about 550 nm for λ seems common, but different sources give very different values for the constant c, ranging from about 1.4 to 1.9.
Why do these constants differ so much? Are they based on different definitions of “optimal” image quality, such as resolution vs. brightness or diffraction vs. geometric blur? Do factors like pinhole thickness or real-world hole shape affect which constant is appropriate?
Originally by Photography Stack Exchange contributor. Source · Licensed CC BY-SA 4.0
Photography Stack Exchange contributor
12y ago
2 Answers
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I can't summarize the whole optical physics theory behind pinhole (mostly because I don't have the proper knowledge!), but I try to explain why there are different values for constant C. One reason that there are different values for C is the fact that one parameter in the calculation of optimum diameter of hole is missing! Let us refer to the wikipedia article that you mentioned:
Within limits, a smaller pinhole (with a thinner surface that the hole goes through) will result in sharper image resolution because the projected circle of confusion at the image plane is practically the same size as the pinhole. An extremely small hole, however, can produce significant diffraction effects and a less clear image due to the wave properties of light.
This means the purpose of C is to find a value that results in good trade off between sharpness and diffraction. Determining this value, however, is dependant on another factor and that is the distance of the subject to the camera.
Circles on the bottom shows the effect of pinhole size on the resulting image.
In the second figure dashed line (geometric limit) is the resolution and solid line is the diffraction. As you can see the diffraction is affected by θ which is a function of distance to the pinhole.
With all being said, IMHO, the whole reason behind different values for C is the fact that it is obtained empirically and each of them had a different value for p (with reference to the first figure).
Copyright
The plots are borrowed from this file. You can find a lot about the pinhole physics in this document.
P.S. I had a look at the source of the mrpinhole.com page and it seems they are using C=1.92.
P.P.S. Having a look at those websites, it seems each of them have a different value for λ and this could lead to different value for C.
P.P.P.S. I agree with MarcinWolny's comment that a perfectly rounded hole is way much more important.
Originally by user11408. Source · Licensed CC BY-SA 4.0
user11408
12y ago
0
Generated from our catalog & community — verify before relying on it.
Yes—different constants come from different assumptions about what “optimal” means and how the blur is modeled.
The usual derivation balances two competing effects:
- geometric blur, which gets smaller as the hole gets smaller
- diffraction blur, which gets worse as the hole gets smaller
A common result is to set geometric blur equal to diffraction blur. For distant subjects, geometric blur is approximated as the pinhole diameter D, and diffraction blur as the Airy disk diameter:
D = 2.44 λ (f/D)
which gives:
D² = 2.44 λf D = 1.562√(λf)
So the often-seen constant 1.56 comes from that specific criterion.
Other constants arise because authors use different definitions of acceptable sharpness or resolving power, different approximations, or different historical derivations. Real pinholes also aren’t ideal: material thickness, edge shape, and hole quality affect performance, so practical calculators may use larger empirical constants. Thicker material can effectively worsen the hole geometry, so real-world “best” values may differ from the ideal thin-pinhole model.
In short: there isn’t one universal constant; it depends on the optimization criterion and how closely the real pinhole matches the ideal model.
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