Resolving power of a diffraction grating For a experiment we have measured the wavelengths of the spectrallines corresponding to a sodium lamp. We have also measured the splitting distance, most likely to happen due to spin-orbit coupling, of a specific spectralline. Accordingly we have calculated the differences in the wavelengths of those splitted spectrallines. Now I was wondering why you are able to calculate the resolving power of the diffraction grating based on these wavelength difference. Is this because the splitting is typically very small and is therefore sensitive to the ability of the diffraction grating to separate closely spaced spectral lines? They do not seem to be anywhere near resolved.

 A: What your image shows are two fringes produced by light approximate wavelengths $589.0\,\rm nm$ and $589.6\,\rm nm$ which is a difference of $0.6\,\rm nm$.
Those two images are quite clearly resolved as shown below at the left-hand side.

I have "moved" the two fringes closer together to show that the fringes could be even closer together and still be resolved.
Analysis of the image on a pixel scale suggests that the resolving power of the diffraction grating, $\lambda/\Delta \lambda$ with $\Delta \lambda \approx 0.13\,\rm nm$ is approximately 4500 and most probably an underestimate.
The closeness of the fringes means that assumption used to find the separation of the fringes, that the displacement in pixels is proportional to the difference wavelength, is a reasonable one.
Theory tells one that the resolving power is equal to $mN$ where $m$ is the order of the fringes and $N$ is the total number of slits illuminated.
Thus measurement of the diameter of the collimator/telescope lens and knowledge of the order of the fringes should enable one get an order of magnitude estimate the number of lines per mm on the diffraction grating.
