Pinhole Design – what Lord Rayleigh really said

Thoughts from two men on pinhole design. Brian Young investigates the relationship between focal length and the diameter of the pinhole and what Lord Rayleigh thought of this.

Writer / Brian Young
Photography / Brian Young, Chris Dreier, Daniel Zrihen

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Introduction

Many pinholers like to construct their own cameras and, with one exception, there is nothing to it as long as you are good with your hands. The exception is, of course, the pinhole itself. To get the best pictures, don’t even think of punching a needle through a piece of tin: buy a professionally cut hole from Lenox Lasers. Bought pinholes come in discrete sizes from 100 micron to 1000 micron in 50 micron steps (500 micron is 0.5mm), thus it is more useful to determine the focal length to suit a particular pinhole size rather than the other way around.

Generally speaking, the smaller the pinhole the sharper the image. However,  if the pinhole gets too small, the image becomes less sharp due to diffraction. The hole must be perfectly circular and clean cut – as with a laser.

If the pinhole is not so small that diffraction is a problem, every diameter (d) has a particular focal length (f) for optimal image sharpness.

Many formulae have been suggested over the years for the ideal relationship between focal length and pinhole diameter but the only one with any scientific credibility is due to the British Nobel Prize winner, Lord Rayleigh.

Lord Rayleigh’s seminal paper, “On Pin-hole Photography” appeared in the Philosophical Magazine in 1891¹ and was the result of more than ten years of work.  The pinholing community seems to be aware of only a fragment of Lord Rayleigh’s  paper and it is the purpose of this article to clarify his findings.

Briefly, a simple version of Lord Rayleigh’s final, and most quoted, formula giving the optimal f-number (N = f/d) for a particular pinhole size when the subject is at infinity is N_R = 505d, when d is in mm. However, that is not the whole story.

Lord Rayleigh’s findings

This is not an appropriate venue for delving deeply into the maths and science but if you would like to know the full details of Lord Rayleigh’s work on pinhole photography, then please download his Collected Works from the net.

Here, we‘ll just look at some of the most important results. The first and, perhaps, most intriguing of these is the fact that if the camera’s f-number is greater than a certain critical value given by N_crit = 910d, a lens will make no improvement in the resolution of the image. To give you some idea of the numbers, with a pinhole diameter of 0.5mm, this critical focal length would be 228mm.

The paper goes on to show that the full formula for the optimal f-number is,

N_opt = 505kd,

where k (= 1 + f/s) and s is the distance of the subject from the camera.

For example, when the subject is at infinity, as in a landscape (in practice, any subject distance over 20 focal lengths can be counted as at infinity.), k = 1; when the subject is one focal length away from the camera, k = 2.

One curious and endearing feature of a pinhole camera is that depth of field is not an issue as it is with lensed cameras. However, it is important to understand that if the camera is designed for a certain subject distance, the camera will only be in focus for distances equal to, or greater than this. Objects between the camera and the subject will not be in focus.

Suppose your pinhole is 0.5mm in diameter and you like to photograph mountains, then your camera should have a focal length of about 126mm (N = N_R). Everything will then be in focus from about 2.5m (20 focal lengths) to true infinity. The same camera will clearly not give good results if you suddenly decide to use it for macro-photography!

For close-up photography with s = f (for example) and using the same pinhole, the focal length would need to be 253mm (N = 2N_R). Now everything will be in focus from about 0.25m to infinity, so you would be able to photograph mountains as well.

As usual, everything comes with a price: the larger the f-number, the longer the exposure required. For the close-up camera, exposures would have to be 3 to 3.5 times longer than with those for the shorter focal length.

Confirming the findings

Every year since 2001, an organisation called World Pinhole Photography Day (or WPPD, for short) has been running an on-line event which invites everyone in the world to submit a pinhole photograph taken on Easter Sunday.²

It started in a small way with only 312 entries but grew to participation of more than 3,700 by 2011, ten years later.

This site should be capable of providing an invaluable source of pinhole photographs against which to test Lord Rayleigh’s findings.

Sadly, not all the photographs (many quite superb) could be used in the study, since very few (about 3%, in fact) came with details from which both focal length and aperture diameter might be determined. In the case of macro-photographs, the situation was even more dire – only one of those examined specified the subject distance.

To assemble a modest collection of 200 photographs which met the criteria, involved looking at all the entries from 2001 to 2005, more than 6,000 images in total.

The photographs were assessed by the author for technical quality. It was found that all the very best pictures (25) had been taken with f-numbers between N_R and 1.7N_R. As quality deteriorated, the number of images with N less than N_R increased. Nearly 60 percent of the poorest quality pictures fell into this category.

Further evidence comes from “Pinhole Photography”, the well-known book by Eric Renner³. On page 129 of the 2^nd Edition are a set of eight images (Figure 5.21a-h) of the same subject shot through different pinholes. Only the last two images are in focus; and these are the only two images for which N was greater than N_R.

To confirm the veracity of Lord Rayleigh’s extended formula which allows for subject distance, the author carried out the following experiment.

Targets, or subjects, consisting of slips of plywood numbered from I to VII were placed on a long board at distances from the pinhole of 100, 200, 300, 500, 700, 1000, and 1300mm respectively. They were then photographed to determine which subjects were in focus and which were not (see the accompanying photograph).

The camera used was the author’s Super Kamera which has a focal length of 154mm and a pinhole diameter of 0.5mm, thus N = 308. Putting these numbers into  the extended formula gives k = 1.22. This means that if Lord Rayleigh is correct, subject distances from about 700mm to infinity will be in focus whereas for distances less than this the subject will be out of focus.

Lord Rayleigh was right: the photograph confirms that the first four targets are, indeed, out of focus. Target I is clearly fuzzy on the photograph and, when the negative is examined through a powerful lens, targets II, III, and IV also show a distinct lack of definition compared with targets VI and VII.

Summary

For optimum technical quality, the f-number for a pinhole camera must be greater than N_R (since the subject distance can never be greater than infinity)._. There is no upper limit for N but, in practice, a limitation is imposed by the size of camera you are prepared to build and the length of exposure you are willing to tolerate.

For subjects at infinity (in practice, more than 20 focal lengths away) use N_R. The closer the subject is to the camera the greater the f-number required (N = 2N_R when s = f; N = 3N_R when s = f/2; N = 10N_R when s = f/9 and so on). Be aware that as the f-number increases so does the exposure time.

The table in the Appendix may be used to determine focal lengths for a given pinhole diameter and subject distance (focal lengths in excess of 1m have been omitted). Entries shown in red are greater than the critical value.

References

1. Strutt, J. W.,  “On Pin-hole Photography”, Phil. Mag., v.31, pp 87-99, 1891; (also Nr 178 of the Collected Works).

2. World Pinhole Photography Day (WPPD), www.pinholeday.org.
3. Renner, E., “Pinhole Photography”, 2^nd Ed., Focal Press, 1999.

 

Appendix

Finding the right focal length, f (units are mm)

d	Subject distance (m)
(mm)	0.25	0.5	1	2	5	10	20	50	100
0.10	5	5	5	5	5	5	5	5	5
0.15	12	12	11	11	11	11	11	11	11
0.20	22	21	21	20	20	20	20	20	20
0.25	36	34	33	32	32	32	32	32	32
0.30	56	50	48	47	46	46	46	45	45
0.35	82	71	66	64	63	62	62	62	62
0.40	119	96	88	84	82	81	81	81	81
0.45	173	129	114	108	104	103	103	102	102
0.50	255	169	144	135	130	128	127	127	126
0.55	393	220	180	165	158	155	154	153	153
0.60	666	286	222	200	189	185	183	182	182
0.65		372	271	239	223	218	216	214	214
0.70		490	329	282	260	254	251	249	248
0.75		658	397	331	301	292	288	286	285
0.80		914	478	385	346	334	329	325	324
0.85			574	446	394	379	372	368	366
0.90			692	514	445	426	418	412	411
0.95			837	590	501	478	466	460	458
1.00				676	562	532	518	510	508

 

Formerly an academic engineer and originally from London, Brian now lives in Berlin where he practices, researches and writes about alternative processes, pinholing and traditional film photography.