A pinhole camera consists of a box in which an image is formed on the film plane which is a distance P from a pinhole of diameter d. The object is at a distance L from the pinhole, and light of wavelength $\lambda$ is used (Fig. 2.66).

Approximately what diameter d of the pinhole will give the best image resolution?

enter image description here

The solution goes something like this :

He says because of geometric optics we see a bright ring of diameter, $\Delta_1$ and it is found using the relation: $\frac{\Delta_1}{L+P} = \frac{d}{L}$.

Then he says because of diffraction we also see a bright airy disk of diameter $\Delta_2$ and $\Delta_2= \frac{\lambda P}{d}$

He then says the sum of diameter is minimum for best resolution and gets $d$.

Doubt: The image will be formed because of diffraction and hence there would be an airy disk but why would image formed by geometric optics is separately considered? (I understood how the diameter due to geometric optics is calculated).

Also, why should the sum of diameters is minimum for best resolution, won't the centers of both disks would same ?(i.e point where the horizontal line intersects the screen). I don't get why the sum of diameters is minimum?

  • $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$ – user4552 Mar 4 '19 at 15:29

Geometric optics is a way to understand the formation of an image by a pinhole: rays that get through the pinhole from the object keep going straight until they hit the film plane. You can also use geometric optics to imagine that a cone of rays will reach the film plane from each point on the object, with the angular width of the cone corresponding to the size of the pinhole and the distance to the object point. That is, geometric optics indicates that each object point gets spread out by a corresponding amount on the film plane. This is a "geometric point spread function", and it is proportional to the size of the pinhole.

Diffraction also produces a point spread function - the Airy disk - but its point spread function is larger when the pinhole is smaller. It is inversely proportional to the size of the pinhole.

So, even though geometric optics indicates that an infinitesimally small pinhole would give nearly perfect resolution, diffractive optics indicates that an infinitesimally small pinhole would give terrible resolution. The problem asks you to find the pinhole size that gives the best resolution: the smallest point spread function.

Two point spread functions

The diffractive PSF is illustrated in the lower left, and the geometric PSF is illustrates at a different scale in the rest of the sketch.

The total point spread function is, roughly, the sum of the two point spread functions. If you plot the sum as a function of pinhole size, there is a minimum at a particular pinhole size. That minimum is the pinhole size that gives optimum resolution.

  • $\begingroup$ Can you put a picture to show the spreads? I still don't get why should the spread be the sum of it. $\endgroup$ – never_mind Mar 4 '19 at 17:58
  • $\begingroup$ I added a drawing. $\endgroup$ – S. McGrew Mar 4 '19 at 23:27

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