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Jeremy AllingtonSmith
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13/09/02
Dispersive astronomical spectroscopy
Jeremy AllingtonSmith,
University of Durham, 29 July 2002
(Copyright Jeremy AllingtonSmith, 2002)
I will start by considering the most usual type of
dispersing element encountered in astronomical
spectroscopy: the ruled plane reflection grating used
in low order. Having used this to establish the basic
principles, I will consider other types of dispersing
element.
1.
Reflection gratings
1.1
Interference condition
As shown in Figure 1, the path difference between
interfering
rays
AB
and
A’B’
is
)
sin
(sin
β
α
+
a
where
a
is the spacing between the
repeated element in the grating from which
reflection (or refraction) occurs.
The interference condition is fulfilled when the path
difference is equal to multiples,
m
, of the
wavelength of the illuminating light. This gives rise
to the
grating equation
:
ρλ
sin
sin
+
=
m
where
a
/
1
=
ρ
is the ruling
density,
m
is the spectral
order and
λ
is the wavelength
of light.
1.2
Dispersion
By differentiating with respect
to the output angle we obtain
the
angular dispersion
m
d
d
cos
=
α
β
AB
A’
B’
a
Figure 1: Interference of light by a
diffraction grating. Although
considered here as a transmission
grating, the same principle applies
to a reflection grating with suitable
care taken with the sign of the
angles.
D
T
f
1
f
2
α
β
ψ
D
1
D
2
Grating
Slit
Detector
Telescope
α
W
s
χ
f
T
Figure 2: Illustration of the principle of a generic grating
spectrograph showing the definition of quantities used in
the text. To be consistent with the grating equation,
α
and
β
have the same sign if they are on the same side of the
grating normal.
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Page 2 of 29
13/09/02
The
linear dispersion
is then
2
cos
f
m
dx
d
d
d
dx
d
ρ
β
λ
=
=
since
dx
d
f
=
2
where
2
f
is the focal length of the camera (see Figure 2).
1.3
Resolving power
In standard textbooks, the
resolving power
,
δλ
≡
R
(where
is the resolution in
wavelength), is usually described as being given by the total number of lines in the
grating multiplied by the spectral order, hence
W
m
R
=
*
(sometimes this is called the
spectral resolution
which can lead to confusion with
).
But
in practice, the resolving power is determined by the width of the image of the slit,
s
,
projected on the detector,
s’
.
Before going further, it is useful to consider the invariance of Etendue in optical systems
(note that fibres and some other optical systems do not conserve Etendue but systems
made from normal optics – mirrors and lenses  do). This is normally stated as
constant
=
Ω
A
n
where
Ω
is the solid angle of radiation incident at a surface of area
A
in a medium with
refractive index
n
. For our purposes, we may set
1
=
n
(since we only consider optics in
air or vacuum) and consider a onedimensional analogue:
constant
'
'
=
=
a
a
ω
where
and
a
are the opening angle of the beam and the aperture dimension respectively.
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This note was uploaded on 07/17/2008 for the course ASTRO 890 taught by Professor Martini during the Spring '08 term at Ohio State.
 Spring '08
 MARTINI
 Astronomical

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