Resolving Power of the Grating

Consider once again the reflection grating in Fig. 2. It is now illuminated with a mixture of light with two wavelengths, $\lambda_1$ and $\lambda_2$.

(a)

Find the positions of the principle maxima (in terms of $\sin\theta$) for each $\lambda$. Sketch (on the same plot) the interference patterns for each $\lambda$. (You can choose any value of $N \geq 3$ for your sketches.) Using the sketches, explain why the grating can be used to separate, or ``resolve'', the different wavelengths.

(b)

The width of a principal maximum can be defined as the distance (in terms of $\sin\theta$) between the two minima neighbouring the maximum. Find the width of the principle maxima for each $\lambda$.

(c)

Consider the situation when the two wavelengths are very close to each other, $\lambda_2=\lambda_1+\delta$. What is the minimal value of $\delta$ for which these waves can still be resolved?

(d)

Sketch (on the same plot) the interference patterns for two wavelengths that are too close to each other to be resolved. (Again, choose any $N \geq 3$.)