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Problem 47.29 (RHK)

A diffraction grating has a resolving power . (a) We have to show that the corresponding frequency range that can just be resolved in given by . From the figure shown, we have to show that thetimes of flightof the two extreme rays differ by an amount ; (c) and that , which is independent of the various grating parameters. We may assume .

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For light frequency and wavelength are related to each other through the speed of light c. We have the relation

From this relation we note that

The resolving for a grating is

We have


The path difference between rays reaching the point P on the screen from two successive rulings of the grating is


If the grating has N rulings, the path difference between the extreme rays will be .

Assuming , we approximate it by .

Therefore, the difference in “times of flight” of the extreme rays will be


We thus find that