A grating with high groove density has a high dispersion. This means that the angular separation of wavelengths from the grating is large. It can be desirable to choose a grating with high groove density for two main reasons:
 To obtain a compact design (for a wide wavelength range)
 To obtain a high resolution (for a narrow wavelength range)
However, there is a physical limit to how high the grating groove density can be. In this post we will derive a simple formula for calculating the maximum groove density G_{MAX} for a grating that should cover a certain wavelength range λ_{MIN} to λ_{MAX}.
The figure above defines the various parameters and also shows that when the groove density reaches G_{MAX}, the largest wavelength is diffracted at 90 degrees which means the order disappears.
The starting point is the grating as written in the

Eq. 1 
First of all, we simplify things by assuming the grating is used in Littrow condition at the center wavelength λ_{C} = 0.5*(λ_{MAX} + λ_{MIN}). This means that the diffraction angle β (λ_{C}) equals the angle of incidence a at this specific wavelength which means:

Eq. 2 
The other condition is that the diffraction angle β equals 90 degrees at λ_{MAX}:

Eq. 3 
By combinig Eq. 2 and Eq. 3 we finally get the following simple and useful formula for the maximum groove density:

Eq. 4 
In reality the groove density needs to be chosen lower that G_{MAX} – typically at 70 – 80% of G_{MAX}
MAY
2018