- #1

JulienB

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## Homework Statement

Hi everybody! Here is the problem I am trying to solve:

a) A source illuminates a grating in a spectroscopical element so that the principal maxima appear as thin bright bands (therefore the name "spectral lines"). Show that the angular width ##\Delta \theta## of such a line is inversely proportional to the width of the grating (in case of normal incidence).

b) Find an expression for the angular width per (small) wavelength's range, i.e. the dispersive power ##\lim\limits_{\Delta \lambda \to 0} (\Delta \Theta/\Delta \lambda)##. Calculate the dispersive power for the 1st and 2nd order of a diffraction grating with a number of 700 line per cm (i.e. ##\lambda \approx 500##nm).

## Homework Equations

Grating equation: ##a (\sin \theta_m - \underbrace{\sin \theta_e}_{=0}) = m \lambda## where ##a## is the width of one slit and ##m=0,1,2...##.

## The Attempt at a Solution

a) We recall that the principal maxima occur for ##\alpha = 0, \pm \pi, \pm 2\pi,...##. The minima occur at ##\alpha= \pm \frac{\pi}{N}, \pm \frac{2 \pi}{N},...## so I calculate ##\Delta \alpha## as the distance between the neighbour minima of a principal maxima, i.e. for example:

##\Delta \alpha = \frac{\pi}{N} - \big(- \frac{\pi}{N} \big) = \frac{2 \pi}{N}##

Recalling that ##\alpha = \frac{k a}{2} \sin \theta##, we can obtain by differentiating ##\Delta \alpha = \frac{k a}{2} \cos \theta_m \Delta \theta## and thus:

##\frac{2 \pi}{N} = \frac{k a}{2} \cos \theta_m \Delta \theta \\

\implies \Delta \theta = \frac{2 \lambda}{N a \cos \theta_m} \\

\implies \Delta \theta \mbox{ is inversely proportional to}\frac{1}{N a}.##

So far so good (hopefully), but now I am stuck at b). I have been searching on internet and in the two books I have but I can't make sense of this limit as it is given in the problem. Neither could I find what this so-called dispersive power is. In the book of Hecht, there is something called the angular dispersion, which is defined as:

##\mathcal{D} = \frac{d \theta}{d \lambda}##.

Could that be the same thing? It looks slightly different though, the limit in the problem is not the definition of a derivative. Could you guys help me understand what I am supposed to do?Thanks a lot in advance, looking forward to reading your answers.Julien.