Spectrograph geometry calculations

Here's a quick walk-through to calculate the geometry of the spectrograph. I guarantee nothing concerning the correctness of what follows (but please do correct me if you find errors).

At the heart of everything is the diffraction equation:

The relevant wavelengths are given by the Raman shifts:

If we're interested in wavenumbers from 150-4000cm⁻¹ and λ₀ = 532nm, the range for λ₁ becomes 536.3-675.8 nm. The edge filter's cut-on wavelength is 540 nm, so the range is in fact 540-675.8 nm - this also means that the spectrometer's lower limit is 278 cm⁻¹.

d is given by the grating in the case of 1200 lp/mm it becomes

Because the CCD is 29.1 mm wide and the focal length of the focusing off-axis parabolic mirror is 152.4 mm we can calculate what angles of the diffracted light we would like. For 540 nm to fall on the edge of the CCD, the angle should be:

The angle for 675.8nm is of course identical except for the sign. It gives us an angular range of ±5.45° around the center angle. The angle of incidence can now be found by solving this set of equations:

Of course not any solution will fit the spectrograph's geometry. The angle of dispersion (γ) for the central ray must match the angle between the off-axis parabolic mirrors as seen from the diffraction grating (2α):

2α is determined by the position of the mirrors and is:

Where lm is the distance between the mirrors. Their diameter is 25.4mm, so lm cannot be shorter than this.

I'm don't think there's a (practical) solution to the equations, but setting lm to 27mm, 2α becomes 10.16° and then these values more or less cover the spectral range:

The CCD then covers the spectrum from 538-680 nm.

For better explanations go here: http://www.astrosurf.com/buil/us/stage/calcul/design_us.htm