Chapter 47: Problem 25

Question: In a particular grating the sodium doublet (Problem 21, Chap 47) is viewed in third order at $\theta_3~=~10.2^o$ to the normal and is barely resolved. Find (a) the ruling spacing and (b) the total width of the grating.

Solution: The difference in wavelngth between the sodium doublet with $\lambda~=~589~{\rm nm}$ is $\delta\lambda~=~0.6~{\rm nm}$. From

$$d\sin\theta_k~=~k\lambda$$

we have that $d~=~9.98~{\rm\mu m}$. Given that the doublet is just resolved at third order means that the condition for resolution

$$\Delta\lambda~=~{\lambda\over N k}$$

gives N = 327.2 lines on the grating, from which we find the total width of the grating to be $w~=~N d~=~ 3.26~{\rm mm}$.