10.5.5 Rayleigh’s Criterion

I believe you have heard of a pin-hole camera before. It is found in every lower secondary science text book. Let me ask you, for the sharpest image, should the pin-hole be tiny, or large? The tinier the better, right? Because if the pin-hole is large, then many rays from one point on the object can arrive at multiple points on the film, resulting in a blurred image. The sharpest image is formed when the pin-hole is so tiny that only one ray from each point on the object arrives at one point in the image. Right? Plot twist. It turns out that, yes, pin holes should be small, but not too small. When it’s too small, the image actually becomes blur again! Why?

Diffraction! When the pin-hole is too small, diffraction of light sets in. So each point object does not form a point image, but rather a smeared blob. Think of the width of the central maximum of the single-slit interference pattern. Based on \displaystyle {{\theta }_{1}}=\frac{\lambda }{b}, the smaller the pin-hole, the larger the smear.

This smearing is of grave concern to microscopy or telescopy because it limits an optical instrument’s resolving power, which is the ability to distinguish two very close objects.

Say we have two point objects A and B. In another universe where light does not diffract, two point images A’ and B’ would have been formed. If A and B have an angular separation of a, A’ and B’ will also be separated angularly by angle a.

In our universe light does diffract. So instead of two point images, we get two smeared blobs. The intensity profile of each blob follows that of a single-slit diffraction pattern: peaking at \theta =0 (at A’ and B’), tapering off on either sides, and reaching zero at the first minimum angle[1] of \displaystyle {{\theta }_{1}}=\frac{\lambda }{b}. If the angular separation a  is comparable to the first minimum angle q1, then we have a concern, because the two smeared blobs will start to overlap. The resultant intensity profile[2] begins to look like one single hump, making it impossible for us to tell whether there is one or two objects.

The generally accepted criterion for whether two images are distinguishable is the Rayleigh criterion, which states that two images are just resolved when the central maximum of one image coincides with the first minimum of the other image. Hence according to the Rayleigh criterion, two images are just resolved when

\displaystyle \alpha ={{\theta }_{1}}=\frac{\lambda }{b}

In any optical system, the area that is receiving the light is acting as the “slit”. For animals, it is the size of the pupil of the eyes that decides the amount of diffraction occurring. That’s why eagles, with their large pupils, have very good eye sight. For satellite dishes, having a larger dish will reduce the amount of diffraction. To have high resolving power to distinguish astronomical objects with very small angular separation, gigantic dishes are built.

Applet

Optical Resolution Model (compadre)

Concept Test

2049


[1] From the single slit formula, the first minimum occurs at \sin {{\theta }_{1}}=\frac{\lambda }{b}  . Since we are dealing with very small values value of q1, \sin {{\theta }_{1}}\approx {{\theta }_{1}}=\frac{\lambda }{b} .

[2] Because the light emitted by the two point objects are incoherent, they do not interfere with each other. We can simply sum up their intensities.

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