...

Longitudinal spatial coherence asks how correlated the E field is between two points along the same direction of propogation. This is essentially just set by the coherence length of the source, which is actually determined by the coherence time of the source and thus is highly related to temporal coherence. For two points that are closer together than the coherence length (c*t_coherence), the E fields are highly correlated. For two points further than the coherence length, the E fields have no correlation.

Van Cittert-Zernike theorem

Suppose we have some extended source emitting light and we have two downstream points P1 (fixed) and P2 (variable) being illuminated.

The mutual degree of coherence between these two points is equivalent to the intensity in the following scenario

A spherical wave converges to P1, passing through an aperture of the same shape and size as the extended source. it diffracts, and we find the complex amplitude at point P2.

Consider the case where the distance to the source is very large compared to the source size. Also assume we have a circular source.

This allows us to approximate the incoming spherical wave as a plane wave. We further approximate that the plane wave has normal incidence.
Then, the mutual degree of coherence between the central point P1 (lying at the 0th order diffraction maxima) and the off axis point P2, 

Diffraction Gratings

Image Added

This schematic here depicts incident light hitting a reflective grating

Applied to spectrometer design

At the grating, we want the incident wavefront to have a transverse spatial coherence equal to or larger than the grating width because this maximizes the number of lines that are illuminated with spatially coherent light. This gives us the maximum spectral resolution since it makes the peaks sharper.

Suppose we have a grating of width 12.7mm. The total distance from the grating to the slit is about 95.4mm. 

Suppose the slit is a circular aperture. What is the minimum slit size needed to