Determining the Maximum Achievable Wavelength of a Monochromator or Spectrograph

A spectrograph mechanism can only rotate the grating through a limited range of angles. The angle and groove density determine the transmitted wavelength. All gratings can be rotated to 0 degrees, so the lowest possible wavelength for a UV grating is set by the transmittance of air at about 180 nm. To determine the theoretical upper wavelength limit of a grating, the following formula is utilized. This formula is often referred to as "the grating equation."

a (sin l + sin D) = mΛ

Where:

m = the order, which is always an integer value

a = the groove spacing or pitch, which is the inverse of the groove density

Λ = the wavelength to be produced by the spectrograph

l = angle of incidence

D = angle of diffraction

In this example, the first order shall be considered, so m = 1. For a 1200 l/mm grating, a = 833 nm. The maximum value of any sine value is 1, so the maximum value of sin l + sin D is 2. Therefore, the longest wavelength achievable in the first order with this grating is 1666 nm. With a 2400 l/mm grating, it the theoretical upper wavelength would be half that of a 1200 l/mm grating. Practical considerations restrict the angles D and L, so the longest usable wavelength is lower than this theoretically possible maximum. Therefore, the upper wavelength limit for this grating may be closer to 1620 nm. This can be considered a "mechanical limit", as it is not related to the efficiency of the grating at the highest achievable wavelength. Although the grating may still be very efficient at this limit, it is the mechanical considerations which prevent the grating from being used at higher wavelengths.

* All mechanical upper limits are theoretical. When calibration is performed, a grating offset is introduced. This may slightly lower the actual upper limit of the grating. Note that the mechanical limit may go to a higher wavelength than the actual usable area of the grating itself.