##### 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.

### 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 1891^{1} 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 a 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 a 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 Renner^{3}. 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:

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

- World Pinhole Photography Day (WPPD), www.pinholeday.org.
- 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 |

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My first reaction was to wonder what self-respecting pinholer would forgo the pleasure and uncertainty of twisting a needle through a bit of aluminum pop can. Where’s the magic in that? But I did learn that focus was not equally sharp/unsharp from the camera to infinity – I have been operating under the assumption that it was.

Being ignorant – or let’s say rusty – in the math involved, it’ll take me more than one pass at the article to digest it and decide how/if it applies to the needle wielding branch of pinholers, but it’s very interesting indeed. Thanks so much for making this freely available to the community. No doubt this will be a very valuable resource for a better understanding of pinhole photography in general and especially for the more technically inclined among us.

Really interesting article. I have the original Rayleigh paper cited here but my knowledge is limited and my English too.

I have a question I would like Brian to consider. One of the formulas relating pinhole diameter with focal lenght and used to calculate the diameter of the pinhole for a particular focal length (or viceversa) which is attributed to Lord Rayleigh is such as pinhole diameter equals a constant (c) multiplied by the square root of the product of focal lenght and the light wavelength (all of the magnitudes measured in mm). My question is related to the “so called constant”. The “usual” value of it is 1.9 (also known as Lord Rayleigh’s value) but you may use a different value for the constant.

I don’t know if this is related to magnification, deep of field or anything else. Is there any relation between the constant and any of the other parameters? If so, could it be explained in “simple” terms?

Thank you very much in advance!

jesusjoglar.net

A good question from Jesus. Lord Rayleigh found the constant, 1.9 by experiment. It is a true, invariable constant and is incorporated, together with the wavelength of yellow-green light, in the number 505 used in the formulae.

The f-number given by 505d is simply the lowest possible value if you want a focused image. The flexibility comes about when you design for a specific subject distance less than infinity using N = 505kd. There is no theoretical upper limit to the f-number but, in practice, the exposure times would become too long.

You might want to write the extended Rayleigh formula as N = Ad where A = 505k. A is now a variable which depends on the subject distance, s. There will be no “depth of field” effects if your subject is more than distance s from the camera. Magnification, which you mention, depends only on the value of f/s.

I hope this answers the question. Good luck.

Thanks for this excellent article!

I have a Finley pinhole camera witch has a bellows that can be extended from 40 – 200 mm and a turret with 4 different pinhole and associated zone plates. 0.23 – 0.34 – 0.46 – 0.54 mm. This is an ideal platform to experiment with Lord Rayleigh findings.

However I have trouble implementing the formula’s and are a bit confused about the table in the appendix.

D in the grey vertical column is the pinhole diameter in mm?

Subject distance in the grey horizontal header row is this the distance from the pinhole or film plane to the subject???? is this in Meters????

Outcome in the white table below is the Aperture (f) ???

Thanks for this question, Rogier.

Yes, the grey vertical column to the left gives the pinhole diameter (d) in millimetres (mm).

The subject distance in metres (m), measured from the pinhole, is given in the grey horizontal row at the top.

The main body of the table gives camera focal lengths (not aperture), measured in millimetres (mm).

I hope that this helps.

Hi Brian,

Thanks for your explanation 🙂

However now I am even more confused. I thought that the focal length of a pinhole camera was determined by the distance from the pinhole to the film. Witch also determined the aperture.

Ah I think the “coin dropped” this is a table to look in reverse to find the optimum focal length (distance pinhole to film).

Rogier

Can anyone help me to calculate the focal length for a 0.23mm pinhole ?-)

26,643 mm

You may use this free software: Pinhole designer v 2.0 by David Balihar. It is very simple to use

and you get it in http://www.pinhole.cz/en/pinholedesigner/

I hope this helps

Jesús

Jesus, I have not had an opportunity to test the link you have given since the calculator is for Windows only. However, I did notice that an allowance for subject distance is not included.

Rogier, to answer your question you can use the formulae given in my article or look at the table below which has been specially determined for a pinhole of diameter 0.23mm. As before, “s” is the subject distance measured from the pinhole and “f” is the focal length (the distance from the pinhole to the film plane):

s (mm) 30 35 40 50 100 200 500 1000

f (mm) 244 113 80 57 36 31 28 27

I hope that this helps.

Thanks for the link but its indeed Windows.

Meantime I have found this link. Also the main website is a great resource 🙂

http://pinhole.stanford.edu/phcalc3.htm