Theoretical
Parameters for the Design of a “Classical” Spectrograph
English translation: Steve Dearden
Version
en francais
The notes below
are for a quick and simple calculation of the geometric and optical parameters
for a classical spectrograph (in contrast to other mountings such as the
Littrow mounting, Grism systems, etc..)
These stepbystep calculations are generalized, and can therefore be
adapted to other classical spectrograph designs.
Click
here for download a Microsoft Excel spreadsheet for a fast calculation
of the this spectrograph and a bonus ... the limit
magnitude estimation !
The basic
spectrograph design that we are going to consider can be seen below:
The following
components are used in this design:
 A 135mm f/2.8 telephoto lens (as the
collimator)
 A 50mm SLR camera lens, f/2 or faster
(as the focusing lens)
 A reflection grating, 30mm square,
600lpm (e.g. Edmund Ref. NT46075)
An optical
schematic of the spectrograph is shown below:
We assume that
the telescope has a principal mirror of diameter (D) equal to 200mm and focal
length (f) equal to 1200mm. The
aperture ratio (f number) is therefore 1200/200 = 6.
D=200 mm f=1200
mm F#=6
Assume also that
the other optical components have the following values:
For
the Collimator: Focal length = f_{1} = 135 mm f ratio = F_{c} = 2.8
For
the Camera Lens: Focal length = f_{2} = 50 mm f ratio = F_{o} = 1.4
For
the Grating: Number of grooves = m = 600 per mm Dimensions = H = 30 mm
STEP 1 – Check for no Vignetting at the Collimator
The collimator
lens will be large enough to pass all the light from the telescope if:
F_{c} < F#
Since 2.8 < 6,
this condition is fully satisfied.
STEP 2 – Calculate diameter d1
d1 is the
diameter of the beam exiting the collimator (refer to the figure below).
Now,
In this case, we
have
STEP 3 – Calculate the angles of incidence and diffraction
of the grating
For the visible,
we want light of wavelength λ_{0} = 5500 angstroms to be centered on the
CCD. This wavelength is known as
the key (or reference) wavelength. We have essentially chosen the centre of the
visible spectrum – the colour green.
Let γ be the total
angle between the incident rays on the grating and the direction of diffracted
for the reference ray (5500 A):
However, due to
the physical size of the collimator and other components, and the need to
minimise the value of the total angle γ, a value of 38° is chosen. This value
is characteristic for the design, and should NOT be increased significantly.
The angle of
incidence a is given by:
where k is the
grating order, which can take values of …2, 1, 0, 1, 2,… In a standard
spectrograph such as ours it is strongly recommended to work in first order (k
= ±1). k = 0 corresponds to the simple
reflection from the grating, and in zero order the grating is acting as a
mirror without dispersing the light. Note that working in zero order allows you
to image the telescope field onto the detector as if the camera was placed
directly at the focal plane, with an image size multiplied roughly by the
coefficient f_{2}/f_{1}.
This ratio is usually smaller than 1, so the spectrograph is behaving like a
simple focal reducing lens. By rotating
the grating it is therefore possible to go from the “spectrographic” mode to
direct imaging, which is very useful for identifying and centring an object
whose spectrum is to be recorded.
The above
equation for a can be solved by rearranging so:
Entering the
numerical values gives:
From which
Thus the two
possible values for α are 29.05° and 8.95° for grating orders 1 and –1
respectively. The corresponding
values for β, using the relation γ = α – β, are β = –8.95° and β = 29.05°.
At this stage, we
choose the solution α = 29.05°; β = 8.95°
(we will see at Step 9 whether this
choice is a good one).
Nb: Angles of opposite sign denote that
the incident ray and the diffracted ray lie on opposite sides of the normal to
the grating.
STEP 4 – Calculate the minimum necessary grating dimension
We
have:
Thus:
Since L < H,
no light is lost by the grating, although the 2mm margin is cutting it fine!
Care and attention must still be paid to correctly align the optical
components.
STEP 5 – Calculate the spectral dispersion on the CCD
Let ρ be the
degree of dispersion, in Å per pixel, on the CCD. For a CCD sensor of pixel size (p) = 9 microns, the dispersion
is given by
which gives
STEP 6 – Calculate diameter d_{2}
d_{2} is
the beam diameter of diffracted rays from the grating, and is given by
where X is the
distance from the grating to the front surface of the camera lens and N is the
number of pixels across the CCD in the direction of dispersion. Owing to
mechanical constraints, X is equal to 60 mm.
Assuming that the CCD is the Kodak KAF0400 type, with N = 768. Therefore
For total absence
of vignetting of the CCD, the condition that F_{0} < f_{2 }/ d_{2}
must be fulfilled. In our case: 1.4 < 1.48. Therefore the
condition is satisfied, but only just. The 50 mm camera lens has to be at full
aperture, which is not the best situation for good image quality and spectral
resolution. An improvement of the
situation is provided in a later section.
It has to be strongly emphasised that the
grating to camera lens distance (X) must be minimized as much as possible in
order to eliminate vignetting and to minimise optical aberrations, whilst at
the same time avoiding blocking rays from the collimator. Obviously, a compromise has to be found
here.
Nb. The ratio is called the
anamorphic factor of the spectrograph (see later).
STEP 7 – Calculate the spectral range covered
The extreme
wavelength limits from one side of the CCD to the other are given by the
relation:
Filling in the
numbers gives λ_{1} = 4363 angstroms and λ_{2} = 6637
angstroms.
Thus the
spectrograph in this configuration practically covers the whole visible region
with an image size of 2.96 A/pixel. The grating must be rotated if the UV or IR
regions are to be examined, so provision to accurately rotate the grating
manually is an important requirement.
STEP 8 – Calculation of the Spectral Resolution
The spectral
resolving power (R) is defined as
where Δλ is the
smallest resolvable spectral element. The greater the value of R, the greater
is the spectral detail. The current spectrograph is of medium resolution with a
value of R around 1000. At this
resolution, spectral lines with a separation of 6 Å in the 6000Å region (eg.
the sodium doublet) can be resolved. We shall see further on if this
performance is achieved.
One expression
for the resolving power for a grating spectrograph is:
where r is the
anamorphic factor defined earlier and FWHM_{t} is the full width at
half maximum of the image of the star on the CCD, in the direction of
dispersion, assuming a monochromatic light source. If the apparent diameter of
the star’s disk is limited (partially blocked) by the slit at the telescope
focus, then FWHM_{t} is the slit width projected onto the plane of
the CCD (or more precisely the Line Spread Function LSF of the slit).
If the
spectrograph is used without a slit (a case known Seeing Limited, meaning
that it is the apparent angular size of the star that determines spectral
resolution), (FWHM)_{t} can be calculated from the following
expression:
where
 φ is the astronomical seeing (in
radians),
 FWHM_{c} is the full width
at half maximum in mm of the image of the point source produced by the
collimator,
 FWHM_{o} is the full width
at half maximum in mm of the image of the point source produced by the
camera lens,
 FWHM_{d} is the full width
at half maximum in mm of the diffracted limited PSF of the grating.
 p is the
pixel size
Note : This definition of FWHM is the value
that one measures in number of pixels plus fractions of pixels in the image
that would then be converted to mm, knowing the pixel size.
The value of FWHMd is given by
This value is very small in our spectrograph
(FWHMd=1 micron). We adopt FWHMd=0 in the example.
If the object
under study is limited by the slit (the Slit Limited case, e.g., when an
object of large angular extent, such as a nebula, is observed), the parameter
(FWHM)_{t} is given by:
where ω is the
slit width at the telescope focus.
Moreover, if (FWHM)_{t} < 2 × pixel size p,
then the spectrum is said to be “undersampled” according to Nyquist theory,
meaning that the final resolution is governed by the pixel size, and in this
case we will have
With standard
photographic lenses and the f ratio of the telescope entering the spectrograph
equal to f/6 (see earlier) we typically have
FWHM_{c} = 10 microns (0.010 mm) and FWHM_{o}
= 15 microns (0.015 mm).
Nb. The camera lens is working at a
greater aperture (lower f number) than the collimator, which gives rise to the
difference in the above FWHM values.
p value for the Kodak KAF0400 CCD chip is 9 microns (0.009 mm).
Assume that the
atmospheric seeing is 4 seconds of arc, a relatively poor value, but one
corresponding to typical skies for amateurs and for long exposures; it also
accounts for telescope guiding errors and flexure. In radians, this gives φ = 2 x 10^{5} rd.
Recalling the
other parameters:
f = 1200 mm, f_{1} = 135 mm, f_{2} = 50 mm,
and the
anamorphic factor, given by
Inputting the
numbers gives
Therefore the
size of a star at 5500 angstroms in the dispersion direction is 19.5 microns,
or about twice the FWHM of 2 pixels. This is a good value since the smallest
resolvable detail must cover at least two sample elements in order to be
detected.
Thus, the final
expression for the spectral resolving power R is:
The theoretical expected
value of around 1000 is almost achieved. Note that this would not have been the
case if the collimator focal length (f_{1}) had been the same as the
camera lens (f_{2 }= 50 mm). A shorter collimator focal length would
have reduced the size and weight of the spectrograph, but then R would have
been reduced to only 625. In other words, the ratio f_{1}/f_{2}
needs to be relatively high (≥ 2) to achieve sufficient resolution.
Note that for a
slitless spectrograph, the key to high resolution is to employ a telescope
with a relatively short focal length. If f = 500 mm if then becomes possible to
use a collimator with f_{1} = 50 mm without appreciable loss in
spectral resolving power under average (4 arcsec) seeing conditions. With modern
apochromatic refractors, the aperture ratio is typically around F#8.
For example I
uses a Takahashi fluorite refractor of diameter D = 128 mm and focal length f =
1040 mm. With F# = 8, the spectrograph’s optical components are working less at
their aperture limits (less aberrations) compared to the above calculated
example. With good quality
Nikon/Olympus/Canon lenses for a beam at F#8:
FWHM_{c} = 8 microns =
0.008 mm (Nikkor 135 mm f/2.8) FWHM_{o} = 12 microns
= 0.012 mm (Nikkor 50 mm f/1.4)
In addition, with
a 128 mm collimator, the astronomical seeing diameter is closer to 3 arcsec.
than 4 arcsec. for typical exposure times. For all calculations made with these
parameters we find FWHM_{t} = 16.0 microns. The apparent diameter of
the star is smaller than two pixels on the CCD, although the effective
resolution is limited by the detector itself. When such undersampling occurs,
the value to adopt for FWHM_{t} is 2 × p = 2 × 9 microns or 18 microns.
The attainable resolving power is therefore equal to 920 (and for this
particular optical configuration it is the maximum possible value given the
pixel size of the CCD chip). There is an advantage in working with a slight
undersampled spectrum – for a spectrograph with a wide slit width the observed
line profile is relatively independent of variations in seeing (atmospheric
turbulence) over ranges of several tens of percent, which is important when
analysis and interpretation of the astrophysical data is made. Avoiding the
necessity of narrow slit widths smaller than 10 microns or so in stellar
spectroscopy is a considerable advantage in efficiency and use of the
spectrograph and is the sole domain of the smalltomedium sized telescopes of
the amateur. (The only real drawback of not using a narrow slit on a small
telescope is that spectral calibration is rendered more difficult, where
unambiguous identification of spectral lines in the acquired spectrum is
required.).
STEP 9 – Choice of Anamorphic Factor
Somewhat
arbitrarily we chose the following angles in Step 3:
α = 29.05° and β = 8.95°.
We will now see
if this was right correct choice with respect to the other possible solution: α
= 8.95° & β = 29.5°.
Let us determine
the image of the star on the CCD and the resolving power with the new
values. Applying the equations used in
Step 8, we obtain the following values:
giving FWHM_{t}
= 20.60 microns and R = 920.
The Full Width at
Half Maximum of the star image on the detector has slightly increased (20.6μ
versus 19.5μ) thereby reducing the resolving power. However, since the
anamorphic factor (r) is now > 1, this has the effect of increasing the
spectral dispersion. Reusing the equation from Step 5, we obtain:
instead of the
value of 2.96 Å/pixel obtained before. This increase in spectral dispersion
compensates for the wider star image (or the wider slit image) and in fact the
latter case with r > 1 produces a slightly higher resolving power than the
case r < 1. This advantage is always
worth using, even though a higher dispersion is not always the final objective
(because high dispersion = lower limiting stellar magnitude possible). Thus, if we want to achieve the highest spectral
resolving power for our example, we should use in the end a configuration with
α = 8.95° and β = 29.05°. (Note that this is NOT a general result – the same
calculations must be performed for every spectrograph configuration that is
envisaged.)
It must also be
emphasised, moreover, that the calculation of d_{2} with the new
parameters (see Step 6) gives 28.2 mm, which is significantly smaller than the
value of 33.7 mm obtained previously. The beam from the grating that enters the
camera lens is less spread, thus the camera aperture required is smaller and
lens aberrations less of a problem. This is an additional advantage in choosing
the last values for α and β.
STEP 10 – Easy Ways to Increase the Resolving Power
In all the
previous calculations the main factor limiting the resolving power is the
physical size of the pixels of the CCD chip. Without modifying or changing
everything in the spectrograph the simplest way to increase the resolving power
is to increase the groove density of the grating. A 1200 lpm grating will more
than double the spectral resolution.
Maintaining the
total angle to 38° (the main geometry of the spectrograph does not change), we
find α = 1.43° and β = 39.43°. Note in
particular that the incident and diffracted rays are now on the same
side of the grating (the signs of the angles are the same).
Going through the
calculations as before, we will see that the grating dimensions are sufficient,
that diameter d_{2} = 25.7 mm, requiring the camera lens to work at
f/1.9 to avoid vignetting, that the spectral dispersion equals 1.16 Å/pixel and
that the resolving power is equal to 1990.
This is fine, but remember also that using a 1200 lpm grating loses
about one magnitude in light sensitivity compared to a grating of 600 lpm. The judicious choice of which grating to
use will depend on the type of observations to be made. Click here
for the computation of the limit magnitude.
STEP 11 – Some Variations
The figure below
shows a variant on the original design that produces similar image quality but
replaces the rather large and heavy telephoto lens with a simple achromat
doublet lens as the collimator. This results in a more compact design than
before. Whilst the distance X is the
same (60 mm), the grating can now be much closer to the front surface of the
collimator since it now takes up much less space. A typical source for the doublet lens is Edmund, with a focal
length of 100 mm and 25 mm in diameter (Ref. E32327). For rays close to or on
the principal optical axis, the image quality produced by the doublet lens is
the same as that of the telephoto lens.
The next diagram
(below) shows what happens when the spectrograph functions in widefield mode,
corresponding to slitless observation of many stars simultaneously. Here the
spectrograph accepts a field of ±3 mm at the telescope focus without any
vignetting (about 0.3Å for a telescope of 1200 mm focal length).
By moving the
collimator closer to the grating lets us image almost the full exit pupil of
the telescope on the entrance pupil of the camera lens, which is an excellent
means of reducing offaxis aberrations. In other words, if we were to place a
screen just in front of the camera lens we would see in this position the sharp
contours of the telescope primary (mirror or objective). In order to carry out this type of
“entrance/exit pupil” imaging, the following conditions are needed:
·
If X’ =
distance between the grating and the front surface of the collimator lens ·
and X =
distance between the grating and the front surface of the camera lens,
the condition must be satisfied approximately.
In this example
we have X’ = 50 mm, X = 60 mm and f_{1} = 100 mm, which is within 10 mm
of the relation and close enough to correctly image the telescope exit pupil in
the plane.
The following
figure is a 3D ray trace of the spectrograph. We can see how the entrance slit
is oriented and how the spectrum is produced on the CCD.
Lastly, the three
figures below are spot diagrams for three wavelengths in the blue, green and
red regions of the spectrum. FWHM of the images is between 15 to 20 microns in
the visible region. These values are acceptable in view of the sampling
requirement for 9 micron pixels – the FWHM image should cover ~2 pixels for
optimum (Nyquist) sampling. Under standard seeing conditions, the spectral
resolution is not significantly degraded, as we have seen earlier.
STEP 12 – Testing
The only
important adjustment to make on the spectrograph is to ensure that the image of
the slit is at infinity as seen by the collimator – in other words, the rays
are parallel before incident on the grating. Using a goodquality finderscope,
focused at infinity, and placed in front of the telephoto (collimator) lens,
adjust the focusing barrel of the telephoto lens until the slit image is in
sharp focus (see photo.).
An entrance slit
is always a important component in a spectrograph. Even a basic design
consisting of two razor blades can be useful. The slit fulfils the following
functions in the spectrograph:
 It isolates the object under study
from its neighbours (relatively large slit widths – few tenths of 1 mm to
several mm – will achieve this).
 It reduces the sky background in
order to increase spectrum contrast (also achieved by wide slits).
 It limits the region to be analysed
in order not to degrade spectral resolution – see Step 8 – useful for
extended objects such as nebulae and comets.
 Adjustments to the spectrograph can
be made in subdued daylight, without saturating the CCD detector.
Point D above is
illustrated below. Our spectrograph is equipped with a micrometeradjustable
variable slit (see photo). This useful
device was obtained from some scrap lab equipment, but anyone with DIY talent
can probably make something similar.
Initial testing
of the spectrograph is best carried out in subdued daylight. In this case, the
spectrograph can be seen mounted at the end of a 120mm refractor (F/D = 8.5):
The target object
is simply a white sheet of paper illuminated either by sunlight or by lamp that
has some emission lines in its spectrum (e.g. a pocket OSRAM fluorescent lamp
has fairly strong Hg emission lines):
The following
examples were obtained with a 1200 lpm grating in the spectrograph. The slit
was closed down to only a few tens of microns. So narrow a entrance slit will
have no effect on the final resolving power, and the measured resolution will
therefore be determined by the intrinsic characteristics of the other
parameters of the system. The refractor used in the test has no direct
influence on the resolution since it is simply playing the role of light
collector from the source. Using the telescope will however ensure that the
beam aperture entering the spectrograph is a known and controlled value. In
this case we have F# = 8.5. It is therefore most important to get as close as
possible to actual observing conditions during the testing so as to objectively
measure the performance of the system.
The image below
shows the spectrograph with a 1200 lpm grating (Egmund Ref. E47077). The total
angle g = 38°. Note that the grating surface is virtually perpendicular to the
optical axis of the collimator in order to disperse the light into the camera
lens:
Some simple
plastic lens covers, drilled with a hole, and a little Araldite, were employed
to make the interfaces between the camera and the slit.
The first
spectrum can be seen below, showing the greenyellow region of the Osram lamp.
Some broad and diffuse emission bands, of little practical use, can be seen,
but also some strong narrow emission lines due to mercury vapour in the lamp:
The 2 Hg lines in
the centre are at wavelengths of 5769.60 and 5789.66 Angstroms (for the
complete spectrum of this lamp, click here). The colouring in the image is
false, but approximates the true colours (RAINBOW command in IRIS software).
All the horizontal lines are due to small irregularities along the edges of the
slit (remember that the slit width is around 10 microns only!) These horizontal
structures are known as the tranversalium.
The spectral
profile below is taken from the spectrum image above. The two Hg lines are
clearly visible in the centre, and are well resolved (horizontal axis is in
pixel number, and the dispersion in this case is 1.1 Å/pixel.
The following
profile is the magnified profile around the two Hg lines. The FWHM is about 1
pixel only, close to the expected value based on the technical choices that
were made in the design. Note that the line profiles or line spread functions
are symmetric without excessive spreading near the baseline, signifying
excellent image quality. Thus for actual observations of the sky, the only
factor affecting the final resolution will be the optical quality of the
telescope and the star’s apparent diameter (for a configuration without too
narrow a slit width).
The next most
useful target for you to familiarise yourself with the functioning of the
spectrograph is diffused sunlight reflected off the white card. The multitude
of Fraunhofer lines allows us to monitor the performance parameters of the
instrument and to learn some of the principal solar spectral features. As an
example, the image below is the solar spectrum around the magnesium triplet in
the green, again with a dispersion of 1.1Å/pixel and a resolving power of 2200
at 5500Å. The wavelengths of these
three lines are 5167.33, 5172.70 and 5183.61Å.
This small
spectral region has a considerable number of lines. It is shown here directly as it comes from the spectrograph. A number of geometric defects are clearly
visible. First of all, the transversalium is not strictly parallel to the
horizontal CCD axis. A spectrum of a star would also exhibit the same degree of
tilt. This problem comes from the fact that the grating grooves are not
perfectly at right angles to the CCD detector axis owing to mechanical
limitations in the design of the homemade grating mount. This defect would
prevent optimal extraction of spectral data from the image and must be
corrected for. The TILT command in IRIS can make this correction, provided that
we know the tilt angle. This can be performed by trial and error, and in this
case leads to a tilt of 0.75°. The spectrum, corrected for tilt, can be seen
below:
Close inspection
of the lines reveals that they are slightly curved, an effect known as “smile”.
This is a very common image distortion effect in spectrographs. This again must
be corrected for, particularly if binning is subsequently used during spectrum
extraction and no loss in resolution is desired. In IRIS, the command SMILE can
be applied, which requires that we provide the radius of curvature of the lines
in pixels. After several successive attempts, again by trial and error, we
obtain a radius of curvature of –7000 pixels.
The final image below shows the result of the SMILE command:
The lines are now
correctly oriented and the actual spectral analysis can begin. The graph below
is the spectral profile of the sun in the neighbourhood of the Mg triplet
exhibiting abundant detail in the numerous lines. The lines are straight and
narrow and undersampling of the spectrum is evident. It is possible to
slightly increase the resolution (about 2030%) by taking a sequence of
multiple spectra and then displacing them slightly each time with respect to
the pixel position, and then applying the “drizzle” technique.
The next spectrum
(below) shows the H (3933.68Å) and K (3968.49Å) lines of calcium in the UV.
This image
clearly illustrates the problem of chromatic aberration with the dioptric
photographic lenses used in the spectrograph. In the UV region, regions of
sharp focus are strongly wavelength dependent. However, everyday consumer camera
lenses are designed to produce sharp images on photographic film in the
visible. The calcium H and Kline region lies well outside their normal field
of application. By turning the focusing barrel of the camera lens it is
possible to achieve focus on the H and K lines, but lines either side of this
region will rapidly become blurred. The problem of chromatic aberration also
exists in the infrared region, but fortunately the wavelength dependency of
focus in the IR is less sensitive, as we shall see below.
The spectrum
below is of the Halpha region in the red (the intense line to the left of
centre). The large band to the right is due to atmospheric oxygen (wavelength
7600Å).
Over the area
covered by this spectrum, which extends to the deep red (between 6230Å and
7100Å), it appears that the Nikon lenses used perform very well and there is no
need to correct for good focus.
Going further
into the IR, the spectrum below shows the sun between 7200Å and 8000Å. The
atmospheric O_{2} band is visible in the centre, and the effects of
chromatic aberration are still fairly weak.
Finally, here is
the spectral profile of the O_{2} band at 7605 Å:
Our spectrograph
is now ready for use and can observe its first real stars! You will find here
an example of a highresolution application, and here, an example of a
lowresolution application.
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