4.1 Random Stray Light
Consider first how much light there is to begin with at
the primary wavelength of interest then compare it to other
wavelengths that may be present as scatter.
4.1.1 Optical
Signal to Noise Ratio in a Spectrometer
To determine the ratio of signal to noise each
of the components must first be quantified.
4.1.2 The
Quantification of Signal, fu
Flux Entering the Instrument (fT):
Ses = area of the entrance slit
= (hw)
BT = total radiance of light
entering the instrument
GA = total illuminated area
of the grating
Then from Eqns. (3-14), (3-11) and (3-12) total flux entering
the instrument is given by:
(4-1)
So to calculate the flux at a given wavelength that will
exit the instrument fu,
let El be
the efficiency of the grating at wavelength l and
Bl the radiance of light
at wavelength l in the focal
plane.
Assume now that the area of the exit slit is perfectly
matched to the image of the entrance slit.
If Sex =
the area of the exit slit = (h'w') (or if a spectrograph,
the total area of
the pixels).
However, there are many cases when the size of the image
of the entrance slit is larger than the exit slit due
to
image aberrations. Light losses of this kind are "geometric
losses" and may be characterized by the transmission
through the system Tg.
Tg =
1 for a perfect system.
The flux at a given wavelength collected by the detector
is given by:
(4-2)
where Tg is
the geometric transmission at wavelength l.
4.1.3 The
Quantification of Stray Light, fd,
and S/N Ratio, fu
/ fd
The luminance of randomly scattered light is proportional
to the flux per unit area on the scattering optic. To calculate
stray light due to random scatter:
Let G = the etendue between the grating and the detector
element.
(4-3)
Let Bd =
the radiance of stray light proportional to the total
flux density fT / GA
C = a factor which expresses the quality
of the optics (including the grating) as a function of
random scatter.
(4-4)
Total scattered flux is proportional to the radiance
of the scattered light, to the area of the entrance
slit, and
the solid angle with which the exit slit perceives
the illuminated optic.
Random flux is given by: fd =
Bd G
then,
(4-5)
and the ratio of flux at the wavelength of interest fu and the random flux fd is:
(4-6)
4.1.4 Optimization
of Signal to Noise Ratio
Optimization requires two things: the maximization of (fu
/ fd) and the elimination of
stray reflections. Taking the terms of Equation (4-6)
in
turn:
C: Obtain the highest
quality optics including a holographic grating if one
is available.
El:
Ensure that the grating is optimized to be most efficient
at the
wavelengths of
interest.
LA2/(hw):
Unfortunately, these may not be totally free parameters
because of dispersion
and bandpass requirements.
Tgl:
The dominant cause of image enlargement perpendicular to
dispersion
is astigmatism.
If present, the height of the exit slit must be enlarged
to collect all available light with subsequent loss in
optical
signal to noise ratio. New aberration correcting
plane gratings for use in certain CZ spectrometers enhance
S/N ratio by significantly reducing astigmatism.
Bl/BT: This
term is the ratio of the brightness at the wavelength of
interest l to the total brightness of the source.; it is not usually a user accessible
function.
4.1.5 Example
of S/N Optimization
This is an exercise in compromise. For example, take a
researcher who owns a 500 mm focal length monochromator
and is dissatisfied
with the signal to noise ratio. Equation 4-6 suggests
that S/N improvement may be achieved by utilizing a longer
focal length instrument; a 1000 mm spectrometer just
happens
to be available. Assuming the bandpass requirement is constant
for both experiments, the groove density, wavelength
optimization,
and size of the grating is the same, then throughput is
halved (from Equation 3-13, all other things being
equal,
etendue will be proportional to the ratio of the focal
lengths).
Optical S/N ratio would be improved by a factor of 2. Referring
to Equation 4-6, the ratio of the squares of the focal
lengths
gives a factor of four and assuming the slit heights remain
the same the slit widths in the 1000 mm focal length system
would produce double the area of the 500 mm system, thereby,
losing a factor of two. The question to resolve
is whether picking up a factor of 2 in S/N ratio
was worth losing half the throughput. In this example,
there
may also be a reduction in the value of Tg,
astigmatism being proportional to the numerical aperture
(which in
this
case would be double that of the 500 mm system).
It is also worth checking the availability of a more sensitive
detector. It is sometimes possible to obtain smaller detectors
with greater sensitivity than larger ones. If this is the
case the total throughput loss may not be as severe as originally
anticipated.
4.2 Directional
Stray Light
4.2.1 Incorrect
Illumination of the Spectrometer
If the optics are overfilled, then a combination of stray
reflections off mirror mounts, screw heads, fluorescence
from anodized castings, etc. may be expected. The solution
is simple: optimize system etendue with well designed entrance
optics and use field lenses to conjugate aperture stops
(pupils). This is achieved by projecting an image of the
aperture stop of the entrance optics via a "field"
lens at the entrance slit onto the aperture stop of the
spectrometer (usually the grating) and then image the grating
onto the aperture stop of the exit optics with a field lens
at the exit slit. This is reviewed in Section 6.
4.2.2 Reentry
Spectra
In some CZ monochromator configurations (especially with
low groove density gratings used in the visible or UV)
a diffracted wavelength other than that on which the instrument
is set may return to the collimating mirror and be reflected
back to the grating where it may be rediffracted and find
its way to the exit slit. If this problem is serious,
a
good solution is to place a mask perpendicular to the grooves
across the center of the grating. The mask should be
the
same height as the slits. If the precise wavelength is
known, it is possible to calculate the exact impact
point on the
grating that the reflected wavelength hits. In this case
the only masking necessary is at that point.
A more common example of this problem is found in many spectrometers
(irrespective of type) when a linear or matrix array is
used as the detector. Reflections back to the grating may
be severe. The solution is to either tilt the array up to
the point that resolution begins to degrade or if the system
is being designed for the first time to work out of plane.
4.2.3 Grating
Ghosts
Classically ruled gratings exhibit ghosts and stray light
that are focused in the dispersion plane and, therefore,
cannot be remedied other than by obtaining a different
grating that displays a cleaner performance. One of the
best solutions
is to employ an ionetched blazed holographic grating
that provides good efficiency at the wavelength of interest
and no ghosts whatsoever. Any remaining stray light is
randomly scattered and not focused.
4.3 S/N Ratio
ant Slit Dimensions
This section reviews the effects of slit dimensions on S/N
ratio for either a continuum or a monochromatic light source
in single or double monochromator. It is assumed that the
entrance and exit slit dimensions are matched.
4.3.1 The
Case for a SINGLE Monochromator and a CONTINUUM Light Source
Variation with Slit Width
Observation: S/N ratio does NOT vary as a function of slit
width.
Explanation: From Equation 3-13 and a review of Section
3, signal throughput increases as the square of the slit
width
(slit width determines the entrance etendue and the bandpass).
Because, the light source is a continuum the increase
in
signal varies directly with both bandpass and etendue.
The "noise signal" also
varies with the square of the slit widths as shown in
Equation 4-5. Consequently,
both the signal and the noise change in the same ratio.
Variation with Slit Height
Observation: S/N ratio varies inversely with slit height.
Explanation: Signal throughput varies linearly with slit
height (from Equation 3-13).
Noise, however, varies as the square of slit height (from
Equation 4-5). Consequently, S/N ratio varies inversely
with slit height.
4.3.2 The
Case for a SINGLE Monochromator and MONOCHROMATIC Light
Variation with Slit Width
Observation: S/N ratio varies inversely with slit width.
Explanation: Signal throughput varies directly with slit
width. (Even though bandpass increases, only the etendue
governs the number of photons available).
The "noise" is proportional to the square of the
slit width. Consequently, S/N ratio is inversely proportional
to the slit width.
Variation with Slit Height
Observation: S/N ratio varies inversely with slit height.
Explanation: Signal throughput varies linearly with slit
height. Noise varies as the square of the slit height.
Consequently,
S/N ratio varies inversely with slit height.
4.3.3 The
Case for a DOUBLE Monochromator and a CONTINUUM Light Source
Variation with Slit Width
Observation: S/N ratio varies inversely with slit width.
Explanation: S/N ratio at the exit of the first monochromator
does not vary with slit width, however, the light now illuminating
the optics of the second monochromator is approximately
monochromatic and the S/N ratio will now vary inversely
with slit width in the second monochromator.
Variation with Slit Height
Observation: S/N ratio varies as the inverse square of slit
height.
Explanation: The S/N ratio varies linearly with slit height
at the exit of the first monochromator. The second monochromator
viewing "monochromatic" light will also change
the S/N ratio inversely with slit height, therefore, the
total variation in S/N ratio at the exit of the second monochromator
will vary as the square of the slit height.
4.3.4 The
Case for a DOUBLE Monochromator and a MONOCHROMATIC Light
Source
Variation with Slit Width
Observation: S/N ratio varies with the inverse square of
the slit width.
Explanation: At the exit of the first monochromator S/N
varies inversely with slit width. The second monochromator
also illuminated by monochromatic light again changes the
S/N ratio inversely with slit width. Consequently, the total
change in S/N ratio is proportional to the inverse square
of the slit width.
Variation with Slit Height
Observation: S/N ratio varies with the inverse square of
slit height.
Explanation: Each of the two monochromators varies the S/N
ratio inversely with slit height so the total variation
in S/N ratio varies as the inverse square of the slit height.
