# Scaling of wavelength range in transmission grating based spectrometers

This note describes how an existing spectrometer wavelength range can be scaled by changing the groove density of the transmission grating, without needing to change other aspects of the design. The starting point is a known spectrometer with a grating with groove density G0, detector width LD, and covering the wavelength range λmax,0 – λmin,0.

The focal length of this spectrometer is given by:

 LF= LDcos (β) G0 (λmax,0 – λmin,0)

where β is the diffraction angle of the center wavelength. The ratio between maximum wavelength and minimum wavelength for a given transmission grating based spectrometer is always a constant, K:

 λmax,0 =K λmin,0

Since we are considering the same spectrometer platform, the detector width LD, diffraction angle β, and focal length LF are constants that cannot change when we change the grating to one with a different groove density. This means that the new grating groove density G1 and start wavelength λmin,1 will be related to the old groove density G0 and old start wavelength λmin,0 as follows:

 G1= λmin,0 G0 λmin,1

The maximum wavelength can furthermore be found as:

 λmax,1 = K λmin,1

## Example

We consider the FREEDOM HR-UV platform covering 190 – 437 nm with a 2420 l/mm grating. If we want to change the grating such that the starting wavelength becomes 250 nm, we can immediately calculate that we would need a grating with groove density 1839 l/mm and the new wavelength range of this spectrometer will be 250 – 575 nm.