# formula for optimal focal length of pinhole camera

Discussion in 'Extreme, Retro, Instant and More' started by michaela_davidova, Sep 13, 2015.

1. ### michaela_davidova

Hello there,
I am building pinhole cameras from any boxes and tins. Usually I calculate my pinhole diameter as I know the focal length and where to put the light-sensitive material, so I use the formula of Lord Rayleigh: d = 1,9 sqrt(f*0,00055), where d = pinhole diameter, f = focal length in mm, 0,00055 = wavelength of day light. Recently I bought micro drills so I already know the exact diameter of the pinhole and I would like to calculate optimal focal length. To me it sounds simple, I would just mathematically change the formula to get result for "f". Something like that: f = (d)^2/0,0019855).
However, I started to search on the internet and I found out that many pinhole photographers use another formula for counting optimal focal length which is probably called Connors formula: f = (d/0,037)^2, therefore formula for optimal pinhole diameter: d = sqrt(f*0,001369). And here is what confuses me!
When I compare Rayleigh´s and Connors´formula the results differ. So my questions are: Are there many formulas for counting optimal pinhole diameter or optimal focal length and it just depends on me which one I choose and which result I prefer and are the formulas I wrote correct? Or in the case I need to know pinhole diameter I use Rayleigh´s formula and in the case I need to know optimal focal length I use Connors´one? Or do I calculate anything wrongly from beginning?
Thank you for your help and time. I would really like to know all the mathematics and physics about pinhole photography I am just not the best in maths..

2. ### dhbebb

Don't know about formulae - empirically. pinhole cameras have been offered with f-stop equivalents of around f160. The actual focal length is more a matter of taste - the wider the angle of course, the more uneven the illumination of the negative. Commercial offerings have usually been 90 to 135 mm. From what I understand of pinhole cameras, sharpness depends to a great degree on making as clean a hole as possible in material which is as thin as possible. I feel your formulae may be an unnecessary complication!

3. ### michaela_davidova

Thank you, David. I am building my own camera. I know size of the pinhole but now I try to figure out how far I should insert my light-sensitive material. Of course, I can experiment and see the result, but it is important for me to understand the principles and mathematics behind.. Anyway, if anybody knows what are the correct formulas or knows some useful links where I can read about it, I will appreciate that. Thank you.

4. ### dhbebb

Michaela, you know what works best for you, but I still can't understand the purpose of the formulae. Most [smaller] purpose-built pinhole cameras use a pinhole to sensitive material distance equivalent to about a 90 mm lens on 4 x 5 sheet film, in other words a distance of 0.6 times the diagonal of the sensitive material. Any other distance is of course possible if you wish. In the case of improvised pinhole cameras (soda cans, trash cans, panel trucks, etc.), the pinhole to sensitive material distance is going to be dictated by the size and shape of the "camera", and the only calculation necessary is to make a pinhole of a size equal to a lens aperture of f160 to 180.
As I noted earlier, making the pinhole as clean as possible and in material which is as thin as possible is the crucial factor for image sharpness (which of course is nothing like lens sharpness). In other words, I feel it logical to first determine the practical size of the sensitive material, then determine the pinhole to sensitive material distance using the formula of 0.6 times the diagonal, and then make a pinhole of appropriate diameter. You seem to be working the other way around!

5. ### glen_h

There is no exact formula.
Diffraction makes the spot bigger as the hole get smaller, so one minimizes the size of the hole plus the size due to diffraction. Depending on how you define the diffraction spot size, the difference is about a factor of two, which agrees with what you show.

6. ### michaela_davidova

Thank you. I use a tin box which I want to convert into the pinhole camera. I want to use 6x9 film in the curved film plane attached with spools inside the box (so I want to calculate the angle of view as well). I bought micro drills and I can drill 0,1, 0,2, 0,3, 0,4 mm.. pinhole so that´s why I try to calculate optimal focal length for the box (the depth is 56 mm).
I found the instruction on Instructables http://www.instructables.com/id/Design-and-Build-your-own-Pinhole-Camera/?ALLSTEPS but when I compared the formulas I am used to use for building the cameras the results differ. Usually I was calculating the pinhole according the focal length and using Lord Rayleigh´s formula but since I know my pinhole size I wanted to know if I can use the same formula but for calculating optimal focal length.
David, I am not sure if I understand your principle of 0,6 times the diagonal. Do you mean that the pinhole was inserted 90 mm from the sheet of film 4x9? Because according to calculation the 90 mm focal length is not 0,6 times of diagonal, or am I missing something? Thank you

7. ### dhbebb

The 0.6 thing ...
As I said, you can position a pin hole at any distance from the film - the sharpness will be the same, the crucial factor is to have a clean hole in thin material with an effective aperture of f160 to 180 or so. However, a popular configuration seems to be 90 mm on 4x5 inch film. The diagonal of 4x5 inches is 162 mm (150 mm is regarded as a standard lens focal length), if we express 90 mm as a ratio of 150 mm we get 0.6. On this principle, if you are working with 6x9 cm, where the standard focal length is 100 to 105 mm, you might like to use a pinhole distance of 0.6 times 105 mm, approx. 63 mm (the normal wide angle lens for 6x9 cm is 65 mm). If you chose this, you might want to make the pinhole size 65/180 mm, or 0.36 mm (I am sure the 0.4 mm drill you have would be fine for this). This to me seems to be all the calculation you need .

8. ### black_silver

http://www.mrpinhole.com/calcpinh.php

9. ### jim_jones|3

The formula for optimum pinhole imaging has been debated for well over a hundred years. I use PinholeDesigner http://www.pinhole.cz/en/pinholedesigner/ with a user constant of 1.4 instead of the Lord Rayleigh constant of 1.9, based on my experience and preference in pinhole images. Others may have different experiences and preferences. For wide angle pinholes, that user constant is sometimes increased to improve image corner sharpness at the cost of some center sharpness. Lord Rayleigh apparently based his formula more on scientific theory than on pinhole photography. Pinhole image sharpness near the optimum pinhole diameter is limited both by geometric optics and wave mechanics, resulting in ambiguity between the two theories. One way to experiment with varying pinhole diameters is to use a 35mm camera with pinholes at several times the normal lens distance and a target that simulates the subjects one prefers. This permits multiple testing on one convenient strip of film with an image in which the pinhole blur is relatively large.
Some of the charm in pinhole photography is its contradiction of hard science. Do your own thing, and let the magic begin!

10. ### donald_miller|5

I just found this as I was thinking of trying some pinhole projects. What I thought of trying and I have not worked any
arithmatic or diagrams yet was starting with around a 15 or 16 quage needle which has a bore comparable to 0.02 inches
and after placement the length would be 0.04 in. This create a field of 45 degrees. I was thinking that having a short light
tube like this would cut down on noise and maybe diffraction. It would also effectively behave like a smaller bore hole in a
paper thin pinhole. Has anyone seen this done before or have any input.

11. ### bruce_jones|5

This gentleman makes quite good pinholes.
http://www.withoutlenses.com/articles/interview/pinhole-is-pinhole

12. ### pasha-X

Most confusing for me is that I get different levels of sharpness for different pinhole sizes.

Turned the bedroom into a camera obscura using sheets of aluminum foil and gaffer's tape. Punched a small hole with a pencil. Intercepted the beam of light with a piece of trans-lum. The image on, what amounts to, the "ground glass" changed sharpness depending on the distance to the pinhole. Therefore, I've empirically demonstrated that there is an "ideal" distance. The big question is: how to determine it using maths? I've run out of time for a big project and would like to have the sharpest possible images (yes, lol, I know I'm talking pinhole, still...). I know I'm grabbing at straws here but would be grateful for any help. Thanks in advance. Aloha

13. ### JDMvW

Here's a brief note on pinhole from Modern Photography

1989-03​

14. ### BeBu Lamar

I would put my bellow on with the pinhole lens cap in front then I have a zoom pin hole.

15. ### murrayatuptown

A ‘walk-in’ camera obscura room has the same phenomenon of a 3D image plane, as the ‘end-illuminated’ cylinder pinhole camera l described.

Even with lenses (4000 mm, f/100 and f/27), viewers inside the room enjoyed the view because the subject was more than a flat field...very 3D, like most photography... it was more like transient video as no images were captured.

16. ### JDMvW

FWIW, here is a picture (turned right side up) of the image in a log cabin camera obscura at New Harmony, Indiana:

17. ### polka

links to two papers I wrote, presenting a mathematical model of pinhole photgraphy. As I am French, they are in french and you have to read french... but there are pictures indeed, and more interestingly, at the end of the second link, two javascripts that let you play with focals, pinhole diameters, etc. These graphic scripts work best with the Microsoft Internet explorer (some scaling problems with other browsers).

les sténopés n'ont pas de cercle d'image, seulement des cercles de confusion (pratique)
les sténopés n'ont pas de cercle d'image, seulement des cercles de confusion

For those interested, another paper on my anamorphic pinhole cameras. Also in french, but on the F295 forum, I posted a lot in english ; look for POLKA !

photographie anamorphique au stenopé

Enjoy, and tell me.

Polka

18. ### murrayatuptown

One of my previous responses for some reason is pending approval. I referenced a Swedish author’s paper on optimal pinhole.

He explains (physicist, IIRC) there are other subjective differences that affect your perception of sharpness (which is on a scale ranging from low to very low). MTF and spatial frequency (don’t need to know or care) affect contrast which affects perceived sharpness...and different subject matter can have different ‘spatial spectral content’... his conclusion is that a slightly smaller pinhole constant generally leans toward better contrast & perceived sharpness...what Jim Jones endorsed.

https://www.kth.se/social/files/542d1388f27654460dee76cf/Pinhole.pdf