The most hyped solar eclipse of the century passed over the U.S. mainland today. I viewed it through a homemade pinhole camera. Pinhole cameras, made of a hole, a length of empty space, and an imaging surface, have only two adjustable parameters: size of the hole, and distance between the hole and the surface, which is a focal length of sorts. How does tuning these two parameters affect our image?
The angle subtended by the sun is about 0.5 degrees in the sky. It’s simple to see that the size of the image of the sun (L) does not depend on the diameter of the pinhole, but rather only on the distance f in the image below:
Figure 1. Pinhole geometry
For a focal length of 1 m, L should be about 9 mm.
A more interesting question is how the image resolution might be affected by tuning these parameters. It might seem that a smaller pinhole should create a sharper image (Figure 2), but there is a limit.
Figure 2. Ray optics picture. A larger aperture yields a blurrier image, since it allows rays from different parts of the object to be mapped onto the same spot on the image.
The smallest spot that can be formed on the imaging plane is ultimately limited by diffraction. By making the aperture too small, diffraction can rapidly blur the resulting image. That effect is illustrated below.
Figure 3. Diffraction introduces uncertainty measurement of the ray’s initial direction.
The resolution of our system is determined by the more dominant of these two effects, so,
which has the following dependence on D and f:
Figure 4a. Making the hole diameter too small is much more detrimental to image quality than making it too large
Figure 4b. Resolution improves with focal length increase.
A contour map showing the resolution as a function of D and f gives a clearer picture:
Figure 5. Contour map of resolution as a function of D and f. Blue is better angular discrimination. Plotted as log(theta in degrees).
Based on the calculations above, the pinhole should have a diameter of around one millimeter, and the focal length should be made as large as allowable.
I placed 3 pinholes, sized approximately 0.8 mm, 0.2 mm, and 0.4 mm, in order, on an aluminum foil. Then light sealed one of our moving boxes (f ~ 1 m) from our recent move. Two hole cut-outs for the eyes allowed me to look inside (and take photos).
The different light levels for the full sun images can be seen in the right panel.
As the eclipse started I whipped out a slightly better camera to document its progress.
Figure 6. Mirror image of the sun beginning its eclipse.
Zoomed in series of images of the eclipse progress in Boston, MA. The most coverage we had was around 65%.
There are several design improvements I would make on the next iteration. First, I underestimated the importance of being able to get very close to the projected image. I would need to place the eye holes closer or baffle in a different way so that the viewer can approach the imaging surface. Relatedly, since pinhole cameras suffer from minimal tilt distortion I would tilt the focal plane so that the image can be viewed more head-on.
See you in 7 years.