Astronomical Spectroscopy - Physics - University of Cincinnati

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could see 8000Å light from first order and 4000Å light from second order at the same time 2 .
An eye would also have to see further into the red and blue than human eyes can manage,
but CCDs typically have sensitivity extending from 3000-10000Å, so this is a real issue, and
is solved by inserting a blocking filter into the beam that excludes unwanted orders, usually
right after the light has passed through the slit.
The angular spread (or dispersion 3 ) of a given order m with wavelength can be found
by differentiating the grating equation:
dθ/dλ = m/(σ cos θ) (2)
for a given angle of incidence i. Note, though, from Equation 1 that m/σ = (sin i+sin θ)/λ,
so
dθ/dλ = (sin i + sin θ)/(λ cosθ) (3)
In the Littrow condition (i = θ), the angular dispersion dθ/dλ is given by:
dθ/dλ = (2/λ) tanθ. (4)
Consider a conventional grating spectrograph. These must be used in low order (m is
typically 1 or 2) to avoid overlapping wavelengths from different orders, as discussed further
below. These spectrographs are designed to be used with a small angle of incidence, i.e., the
light comes into and leaves the grating almost normal to the grating) and the only way of
achieving high dispersion is by using a large number of groves per mm (i.e., σ is small in
Equation 2). (A practical limit is roughly 1800 grooves per mm, as beyond this polarization
effects limit the efficiency of the grating.) Note from the above that m/σ = 2 sin θ/λ in
the Littrow condition. So, if the angle of incidence is very low, tanθ ∼ sin θ ∼ θ, and the
angular dispersion dθ/dλ ∼ m/σ. If m must be small to avoid overlapping orders, then the
only way of increasing the dispersion is to decrease σ; i.e., use a larger number of grooves
per mm. Alternatively, if the angle of incidence is very high, one can achieve high dispersion
with a low number of groves per mm by operating in a high order. This is indeed how echelle
spectrographs are designed to work, with typically tanθ ∼ 2 or greater. A typical echelle
grating might have ∼ 80 grooves/mm, so, σ ∼ 25λ or so for visible light. The order m must
be of order 50. Echelle spectrographs can get away with this because they cross-disperse
2 This is because of the basics of interference: if the extra path length is any integer multiple of a given
wavelength, constructive interference occurs.
3 Although we derive the true dispersion here, the characteristics of a grating used in a particular spectrograph
usually describe this quantity in terms of the “reciprocal dispersion”, i.e., a certain number of Å per
mm or Å per pixel. Confusingly, some refer to this as the dispersion rather than the reciprocal dispersion.