# Explanations and equations

I realize I’ve forgotten to explain some of the decisions I’ve made in the design process.

The diagram for the spectrograph looks like this:

The angle of incidence (θi) of the collimated light from the slit and the angle of the diffracted light (θm) is determined by the grating equation:

d(sin θi + sin θm) = mλ

where m is the diffraction order (I’m interested in first order diffraction, so m=1) and d is the slit separation. The sign of θi is negative if it’s on the opposite side of the grating normal compared to θm.

The diffracted light of interest is from 500-4000 cm⁻¹ in the Raman spectrum, which with an excitation wavelength of 532 nm translates to 546.5-675.6 nm, because:

$\bar{\nu{}}=\frac{1}{\lambda{}_0}-\frac{1}{\lambda{}_1}$

Since I’m only interested in this small part of the spectrum, it’d be a waste of potential spectral resolution to record anything outside of it. The focal length of the camera lens should thus be chosen so it images only this on the CCD.

Of course there’s more than one parameter available for adjustment. The angle of incidence of the collimated light will be constant in my spectrometer, but that doesn’t mean I can’t decide what it’s going to be.

With varying θi the range of θm also changes. You can probably set up an equation to find the perfect angle of incidence for the components of your spectrometer, but I simply made a spreadsheet¹ that calculates the optimal focal length as a function of θi.

Dv is the difference between θi and θm and it decides the angle between the lens tube with the the slit and collimator, and the camera lens and CCD.

¹ Look at sheet2 in the spreadsheet.