24Mar/100

## MS257 output divergence

the MS257's spec sheet calls out the input F/# in order to match the source to the monochrometer however it doesn't specify the output F/# or divergence which is extremely important to couple the monochromatic light into an optical system.  The spec sheet does instead specify the input and output focal lengths which we can use to back calculate the output divergence.
We first must find the F/# at the output, which is only possible if we assume the input and output apertures are the same. This is a reasonable assumption for a monochrometer as the instrument requires matching input and output slits to function properly. Using the definition of F/# we first solve for d:

$F/\#=\frac{fl}{d}\;\therefore \;d=\frac{fl}{F/\#}$

Once we have this standard equation, we plug it back into the F/# equation with the different focal lengths:

${F/\#}_o=\frac{fl_o}{d}=\frac{fl_o}{\left(\frac{fl_i}{{F/\#}_i}\right)}$

This equation can be rewritten to show that the ratio of the output F/# to the input F/# is equal to the ratio of the output focal length to the input focal length.

Now that we have a close form solution, we can plug-in the constants - the MS257 specification states the instrument has an input focal length of 220mm and an output focal length of 257.4mm; when we plug those values into the equation with the original input F/3.9 we get an output of F/4.56, and by using the equation from the previous post, that translates to a divergence of 12.51° from the exit slit.