I was working thru a derivation of the resolving power of a diffraction grating at the $n$th order. Here is that derivation:
Let us say we have a diffraction grating width $W$ and are looking at the $n$th order. The light from the diffraction grating is of width $Wcos(\theta)$ where $\theta$ is the angle to the normal that the $n$th order makes. The $n$th order occurs when $n\lambda=dsin(\theta)$ differentiating this we get $n\delta \lambda=d\delta \theta cos(\theta)$. Subbing in $\delta \theta=\frac{\lambda}{W cos(\theta)}$ which is the resolution from a single slit and with rearangment we get resolving power $=\lambda/ \delta \lambda=nN$. Where $N$ is the total number of slits.
My question is why have we used the minimum resolvable angle from a single slit? Here are two of my thoughts:
- The beam takes the shape of a single slit and therefore will propagate as if it had passed thru a single slit even though it has not. (this does not account for the 'gaps' in the beam though i.e. when there are no 'holes' in the grating). So what I am saying is that the a beam of any shape will spread out as if it has been diffracted by a aperture the shape of that beam (is this right?).
- We are viewing the pattern with some sort of aperture and it is this that we are recording (I think this is unlikely as we could equally just view it on a screen and we would have to use the diameter of the aperture for this and not that of the beam).