Pinhole Camera Designer

A pinhole camera has no glass to grind and no focus to miss. It is a dark box with a hole at one end and something sensitive to light at the other. That is the whole machine.

How a hole draws a picture

Light travels in straight lines. Every point of a sunlit scene throws light in all directions, but only one thin pencil of it finds the hole, and that pencil carries on to a single patch on the far wall. The patch from the top of a tree lands low; the patch from its roots lands high. Add up every patch and the wall carries the whole scene, upside down and backward, with no lens involved.

Because nothing is being focused, nothing can be out of focus. Near and far objects are drawn with the same softness, which is why the depth of field of such a camera is described as basically infinite. The softness itself is the price. Each patch is at least as wide as the hole that made it.

So the first instinct is obvious. Make the hole smaller.

Why smaller stops helping

It works, for a while. Halve the hole and each patch halves with it, and detail that was smeared together comes apart. Up to a point the rule is as flat as it sounds: the smaller the hole, the sharper the image, and the dimmer. For someone holding a sewing needle and a square of kitchen foil, that is sound advice.

Then light stops cooperating. Light is a wave, and a wave squeezed through a small round opening does not come out as a neat pencil. It spreads into a bright central region, known as the Airy disc, with a series of concentric rings around it. An optical system whose resolution is limited only by this spreading, and not by its own flaws, is called diffraction limited, and no opening of that size can do better.

The awkward part is the direction of the effect. The far-field formula puts the radius of that disc at 1.22 times the wavelength, times the distance to the film, divided by the diameter of the hole. The diameter sits underneath. A smaller hole therefore throws a larger disc, and a hole made half as wide to halve one blur doubles the other.

Two blurs, then, pulling opposite ways. One is the hole’s own width, which shrinks as the hole shrinks. The other is the diffraction disc, which swells. Plot both against the diameter of the hole and they cross like the blades of scissors. To the left of the crossing diffraction rules and the picture goes soft; to the right the hole itself is the blur. Somewhere near the crossing lies the sharpest picture the box can make.

One formula, four constants

Every published rule for that best size has the same shape: the diameter equals a constant, times the square root of the wavelength multiplied by the distance from hole to film. The square root is the useful surprise. A camera four times as deep wants a hole only twice as wide. For the wavelength, writers usually take 550 nanometers, the yellow-green middle of the visible range, close to the 555 nanometers at which the eye is most sensitive.

The constant is where the sources part company. Joseph Petzval, in 1857, added the two blurs and looked for the smallest total, which gives the square root of two, about 1.41. Lord Rayleigh brought wave theory and photographic experiments to the question and judged the sharpest images to come from a larger hole. A paper in the research journal of the United States standards institute writes his result as the square root of 3.6, about 1.90, while his own 1891 lecture gives 2. A third rule, credited to work by Young in 1971, uses the far-field pattern and lands at 1.56, the size at which the hole is exactly as wide as its own Airy disc.

So the published constants run from 1.41 to 2, and none of them is an error. They answer slightly different questions about what sharpest means. The standards paper, having computed the diffraction patterns in full, found their minima shallow and called the difference between Petzval’s value and Rayleigh’s insignificant for practical purposes. It also repeats an observation from working photographers: best diameters are usually quoted to 0.01 mm, yet deviations on the order of 0.1 mm have little effect on the picture.

Take a box 100 mm deep and yellow-green light. Wavelength times depth is 0.055 square millimeters, and its square root is 0.2345 mm. Multiply by 1.56 and the hole should be 0.366 mm across; Petzval’s constant gives 0.332 mm and Rayleigh’s lecture 0.469 mm. Set the tool above to 100 mm and it shows the same figures, with its two curves crossing at 0.366 mm. At that size each blur is 0.366 mm wide, and a rule-of-thumb combination of the two comes to about half a millimeter.

That half millimeter deserves a hard look. Across a sheet 127 mm wide it amounts to roughly 245 blur widths, and no choice of hole improves on it by much. More detail needs a bigger sheet in a deeper box. The figure is also an estimate and not a measurement. Adding two blurs as a root sum of squares is a convenient habit, while the real pattern of light behind a pinhole is more intricate than either simple curve, which is partly why careful analyses disagree about the constant in the first place.

Apertures in the hundreds

Divide the depth of the camera by the diameter of the hole and the result is its f-number, the same quantity engraved on a lens barrel. The example camera works out at 100 divided by 0.366, or f/273. A film maker’s old pamphlet on building pinhole cameras recommended a hole about 1/75 inch across, made by pushing a No. 10 sewing needle halfway up its shank, and its cover picture was taken with the film 4 1/2 inches behind such a hole. That is roughly f/340.

Numbers like these are far beyond the scale of any light meter, but the conversion is plain arithmetic. The light admitted follows the area of the opening, so halving the f-number admits four times as much, and exposure time scales with the square of the ratio between two f-numbers. Suppose a meter reads 1/125 second at f/16. The ratio of 273 to 16 is about 17, its square is about 291, and 291 divided by 125 gives 2.33 seconds.

Two seconds in bright sunlight is also what the pamphlet recorded for its cover picture. Shade, interiors and slower materials stretch that quickly. The pamphlet’s own indoor picture, lit by a single photolamp ten feet away, took 5 minutes.

Paper is slower still. A lesson plan published by one city government gives a starting table for photographic paper in a cigar box camera working at f/250: 49 seconds in bright sun, 2 minutes in weak or hazy sun, 14 minutes in open shade or under heavy overcast, and 38 minutes at dawn or dusk.

Long exposures meet a second complication, and this one lives in the film. In principle a material responds to total exposure, so half the light for twice the time should leave the same density. In practice, at low light levels, the image forms less efficiently, and the same total exposure spread over a longer time gives a thinner negative. The effect is called low intensity reciprocity failure. It generally becomes significant at exposures longer than about one second for film, and above 30 seconds for photographic paper.

Beyond that, the correction belongs to the emulsion and to nothing else. A simple model raises the metered time in seconds to a power. Typical exponents run from 1.25 to 1.45, though some are as low as 1.1 and others as high as 1.8; with an exponent of 1.31, ten metered seconds become about twenty. The spread shows how far films differ. For a metered 100 seconds, an exponent of 1.1 asks for 158 and an exponent of 1.8 for nearly 4,000.

The first measured rule had the same shape. Around 1900 the astronomer Karl Schwarzschild, comparing star images on his plates, found the same blackening whenever the intensity multiplied by the exposure time raised to the power 0.86 came out equal. Counted from one second, that is an exponent of about 1.16, and it turns a metered 100 seconds into about 212. His figure proved to be of limited use beyond his own plates. No single exponent spans the gap, which is why a calculator should take the figure from the published data for the film in the camera and never supply one of its own.

Photographic paper is the gentler way in. It can be loaded under a safelight where most film needs total darkness, it has long been sold in sizes such as 4 by 5 and 5 by 7 inches, and the negative it produces can be contact printed to make a positive: laid face to face on a fresh sheet, pressed flat under a piece of glass and given a second or two of white light. The cost is speed: the pamphlet warned that paper would probably need about 2 minutes for a sunlit subject where film might need 1 or 2 seconds.

After that the method is trial and correction. The pamphlet asks for a paper negative a little darker than an ordinary print. The lesson plan is blunter: if the sheet comes out blank or very faint, try again with double the time, and if it comes out very dark, cut the time in half.

The look, and the limits

A pinhole picture darkens toward its edges, and the wider the view, the stronger the effect. The reason is plain geometry. Light that reaches the film at an angle has traveled farther, it sees the opening as a narrower ellipse, and it strikes the surface obliquely. The longer path costs the square of the cosine of the angle and the other two cost one cosine each, so the illumination falls by the fourth power of the cosine, the rule known as the cosine fourth law. At 30 degrees off the axis the loss is 0.83 of a stop; at 45 degrees it is exactly 2 stops. A short box behind a large sheet easily pushes its corners past that.

The plate matters as much as the hole. The best pinhole is perfectly round, because irregular edges add diffraction of their own, and it sits in extremely thin material. As the diameter of the hole approaches the thickness of the metal, the sides of the hole obstruct light arriving at anything but a right angle, and the corners darken further. Hence the usual craft: thin metal sanded thinner, a needle held square to the surface, the burr sanded off the back.

And the box must be dark. Both sets of instructions agree on the essentials: blacken the inside and keep the hole covered between exposures. The pamphlet adds tape along every edge, and the lesson plan names a light leak first among the causes of a sheet that comes out black. A long exposure gives a stray leak just as long to fog the sheet.

None of this makes a pinhole camera a precise instrument, and that is rather the point. The arithmetic puts the hole within a tenth of a millimeter of its best size and the exposure within reach of a sensible first guess, which is as close as the physics rewards. The rest is a firm support, a patient count, and the pamphlet’s own habit of making three exposures of every scene: one at the estimate, one at double and one at half.