Stray light and the effect it
has on Optical Signal to Noise ratio (S/N) falls into one
of two major categories either a) random scatter from mirrors,
gratings, etc. or b) directional stray light such as reflections,
reentry spectra, grating ghosts and grating generated
focused stray light.
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. 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. (From Eqn. (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 for the researcher 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 ion etched blazed holographic grating
that provides good efficiency at the wavelength of interest and no ghosts
whatsoever. What stray light may be present 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 Eqn. (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.
