Question

A shoebox with a tiny hole in it (instead of a lens) can be used to take photographs, the main disadvantage being the long exposure time required.

Consider a camera with a hole of diameter d and with the film a distance L behind the hole. An infinitely distant point source (e.g., a star) would ideally produce an image that is point-like. In this problem, you will determine how closely one can approach this ideal. First, using geometrical optics, find the diameter of the image on the film. Next, taking into account diffraction by a circular aperture, find the diameter of the Airy disk on the film. Each of these two estimates gives a rough lower bound on the actual size of the image. Sketch the shape of these two estimates as a function of d. For approximately what value of d do you get the smallest image? If your shoebox has a depth of 0.5 meter, estimate the minimum diameter of the image attainable.

Answer 1

Points to remember

1. If diameter of pinhole camera is too large the picture will be blurred as more number of light rays will form numerous images on the screen. (light ray)

2. If diameter of pinhole camera is too small the picture will be also blurred. This is because diffraction of the light. An airy disc is formed around the geometrical image on the wall.(wave nature of light).

• For infinite object, incident parallel rays, using the ray optics the image diameter will be same as pinhole diameter (d).
• If we consider the diffraction effect the diameter of the first airy disc on the film (wall) will be,

, where d is diameter of the hole, L is distance from pinhole to film, is wavelength of ligt (say for yellow light 580 nm).

Now to get the smallest image (best resolution), the condition to be satisfied is,

or,

See this image for visual clarification,

using L=0.5 m, and wavelength=580 nm=580*10-9 m

d=0.8484 mm

So, the smallest size of the image will be formed when pinhole diameter is 0.8484 mm.

So the minimum diameter of the image will also be 0.8484 mm.