We first consider the relation:
$$n\delta{\lambda} = d\delta{\theta}\cos{\theta}$$
It's content is that the $n^{th}$ order maximum of a wavelength $\lambda + \delta{\lambda}$ is displaced from the corresponding maximum for a wavelength $\lambda$ by the angle $\delta{\theta}$, related to $\delta{\lambda}$ by the above equation.

Now, we can ask the question, "for what (minimum) value of $\delta{\lambda}$ can we clearly distinguish between the $n^{th}$ order maxima of $\lambda$ and $\lambda + \delta{\lambda}$?" The answer is that we can certainly do this (using the Rayleigh criterion) when the angular width $\delta{\phi}$ of the $n^{th}$ order maximum of light of wavelength $\lambda$, on either side of the maximum, is less than the separation of the maxima, $\delta{\theta}$ i.e. when
$$\delta{\phi} \le \delta{\theta}$$
or, the minimum value of $\delta{\lambda}$ that can be just resolved is one for which
$$\delta{\phi} = \delta{\theta}$$

Now, what about the spread $\delta{\phi}$ of the $n^{th}$ order maxmimum? When considering the grating as a series of a large number of slits $N \gg 1$ with separation $d\cos{\theta}$, you can see that a minimum occurs at an angle for which the contribution from a slit of position $m \le \frac{N}{2}$ is out of phase with that of position $m + \frac{N}{2}$, so that each of these pairs have a net zero contribution (note that we can always consider $N$ to be even, when it is large, by neglecting the contribution from one slit if necessary). Therefore, with the diffraction grating width of $W = Nd$, we see that the required criterion is that slits at a separation of $\frac{W\cos{\theta}}{2}$ are out of phase i.e. that (for the first minima from the centre)
$$\frac{2\pi}{\lambda} \frac{W\cos{\theta}}{2} \delta{\phi} = \pi$$
$$\delta{\phi} = \frac{\lambda}{W\cos{\theta}}$$
Note that a similar argument can be used for diffraction from a single, continuous wide slit (which is comparable to this case as both deal with a large number of point sources).

Thus, we now have, on equating $\delta{\phi}$ and $\delta{\theta}$,
$$\delta{\lambda} = \frac{\lambda d}{nW} = \frac{\lambda}{nN} \implies \frac{\lambda}{\delta{\lambda}} = nN$$

This result is independent of your methods of observation (aperture or otherwise) so long as you take care to observe all parallel rays inclined at an angle $\theta$, focused at a point.

## Best Answer

A diffraction grating is a device with a spatially varying phase. It may look like a series of slits but it's more like a bunch of lines carved into glass so that at different points the light sees a different amount of glass and thus different phase. It can transmit both vertical and linear polarization, though they won't be diffracted the same. The width of the lines and the wavelength of the light influences the angle of diffraction.

A polarizer is a device that selectively attenuates one polarization. There are wire grid polarizers just as you've said that absorb light in one polarization in order to achieve this. I don't know to about their construction, but I imagine that the slit width would provide a trade-off between how well you've polarized your light and how much light is left after it.