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

Gλ = sin(α) + sin(β) | 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:

Gλ_{C} = 2sin(α) | Eq. 2 |

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

Gλ_{MAX} = sin(α) + 1 | Eq. 3 |

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

G_{MAX} = 1 = 4 λ_{MAX} – ½λ_{C} 3λ_{MAX} – λ_{MIN} | Eq. 4 |

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

[callout font_size=”12,5px” width=”500″ style=”royalblue”] As an example let us look at a grating for the 800 -1100 nm wavelength range. From Eq. 4 we can quickly calculate that the maximum groove density is 1600 lines per mm. This means, that a practical grating for this range should not have more than approximately 1500 lines per mm.[/callout]