Pinhole Photography: Pinhole Diameter Calculation, Exposure Time Conversion, and Image Field Range

1. The physical nature and geometric optical boundaries of pinhole imaging

Pinhole photography completely abandons all complicated and thick glass lenses and only relies on the oldest principle of linear propagation of light in geometric optics for imaging:

1. Counter-intuitive optical infinite depth of field:

• Since there is no refractive lens to focus, the light passes directly through the tiny pinhole and is projected onto the photosensitive film;
• Pinhole cameras have no physical focal length and focusing mechanism! From a surface-to-surface object 1 centimeter away from the pinhole to a distant galaxy tens of millions of kilometers away, the entire panorama is in absolutely equal focus sharpness.

2. Dilemma: Geometric dispersion vs optical diffraction:

• The pinhole is too thick (such as 2mm): the geometric light spot is too large, the light overlaps seriously, and the picture is blurry like a large frosted glass;
• The pinhole is too thin (such as 0.05mm): According to the wave theory of light, light waves undergo violent Fraunhofer Diffraction at the edge of extremely small apertures. The light is scattered and scattered, and the picture is also violently blurred and the exposure time is as long as several hours.

2. Rayleigh criterion optimal pinhole diameter calculation formula (Rayleigh Optimum)

Physicist John William Strutt (3rd Baron Rayleigh) gave a mathematical formula for the optimal aperture that takes into account geometric clarity and minimal diffraction:
$$D = c \times \sqrt{\lambda \times f}$$
Among them:

• $D$: optimal pinhole diameter (mm mm);
• $f$: the physical distance from the pinhole to the photosensitive film (equivalent focal length, mm);
• $\lambda$: central wavelength of visible light (usually yellow-green light $\lambda \approx 0.00055 \text{ mm}$, that is, 550nm);
• $c$: constant constant (usually $c \approx 1.9$ recommended by Lord Rayleigh or $1.41$ recommended by Petzval).

Quick lookup table for optimal pinhole diameters for common focal lengths:

• Ultra wide-angle end (focal length 35mm): The optimal pinhole diameter is 0.26mm; the aperture value is approximately f/135.
• Standard end (focal length 50mm): Optimum pinhole diameter is 0.31 mm; aperture value is approximately f/160.
• Telephoto end (focal length 100mm): The optimal pinhole diameter is 0.44mm; the aperture value is approximately f/227.

3. Hand-made ultra-high-precision laser-grade pinholes (brass sheet grinding method)

Ordinary sewing needles directly punch holes in Coke cans, and the hole walls will have burrs and thick shadows, resulting in black edges and severe dispersion:

1. Material selection (0.05mm ultra-thin copper/brass sheet):

• The thinner the better, to prevent the pinhole from turning into a "micro-pipe" and blocking the slanted light from the edge.

2. Sewing needle rotation and spin engraving method:

• Place a thin copper sheet on a hard board;
• Use the finest No. 00 insect needle or a very fine sewing needle, press it vertically on the center of the copper piece, gently rotate the needle body with both hands, and press out a tiny bulge on the back of the copper piece;
• Turn over the copper sheet and use 1500 mesh ~ 2000 mesh ultra-fine water-resistant sandpaper to grind the raised areas horizontally to thin the copper sheet;
• Gently turn the needle tip again to penetrate the copper.

3. Eliminate burrs and optical blackening:

• Check under a high-power magnifying glass or microscope: the pinhole must show perfect roundness without any metal burrs on the edge;
• Use matches to smoke or spray matte high-temperature resistant matte black paint to blacken the copper holes to completely eliminate the reflection on the metal surface.

4. f/200 ultra-small aperture exposure calculation and reciprocity law failure compensation

Since the pinhole aperture is usually as small as f/180 or f/250, it cannot be read by any in-camera automatic light meter:

1. Basic conversion method (convert f/22 to f/180 on a light meter):

• First use an ordinary digital camera or a handheld light meter to measure the shutter time required for the f/22 aperture at ISO 100 (for example, f/22 corresponds to 1/4 second under sunny daylight).
• Going from f/22 to f/180 is a full 6 EV stops ($2^6 = 64$ times).
• Theoretical pinhole exposure time: $1/4 \text{s} \times 64 = 16 \text{ seconds}$.

2. Reciprocity Failure of superimposed film long exposure:

• The emulsion efficiency of the film drops sharply under extremely slow exposure of more than 1 second, and a large amount of exposure compensation must be added!
• Using regular Kodak Tri-X 400 film as an example:
• Theoretical calculation time 16 seconds $\to$ actual exposure must be as long as 45 seconds to 60 seconds!

3. Visual poetry of the finished film:

• The extremely long exposure time turns the trees swaying in the wind into hazy smoke, and completely erases the turbulent crowd into nothingness, leaving only the eternally still buildings and soft and dreamy surreal light and shadow, exuding the original charm of 19th-century classical photography.

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