*The catalog lenses are for illustration only.
How To Optimize Spectrometer Light
Throughput for Maximum Sensitivity
A.Introduction
Simply put, we have a sample area that reflects
transmits or emits light that must be projected
through the entrance slit of the spectrometer with
minimal losses.
We use Geometric etendue (geometric extent), G, to
characterize the ability of an optical system to
accept light. It is a function of the area, S, of the
emitting, reflecting or transmitting source and the
solid angle, (Ω), into which it propagates.
Etendue therefore, is a limiting function of system
throughput. The optic with the lowest G value
creates the light-throughput bottle-neck.
While not a geometric factor, light throughput also
depends on the characteristics of the sample
spectra, and is not included in etendue calculations.
Light throughput is affected by whether light is
narrow versus band radiation.
B.Know the spectral characteristics of your light
source spectra
1.
Line spectra such as atomic emission: For
example, elemental excitation, astronomy, ion-
discharge lamps …
2.
Continuum spectra typical in hyperspectral
imaging: The light source emits a continuous
spectrum without a break, such as a halogen
lamp. Applications include remote Earth sensing
used by the USGS, USEPA, USDA, color
measurement, analytical laboratories,
forensics…
3.
Hybrid such as fluorescence, LED, OLED,
where a bandpass can be considered a
continuum over a short wavelength range.
C.The impact on light throughput as a function
of spectral “type”
1.
For a continuum light source: Light throughput
increases with the square of the slit width.
(There are available photons with increasing
bandpass).
2.
For line emission spectra: Light throughput
increases linearly with slit width. (There are a
fixed number of photons in a particular emission
line. There are no additional photons in the
source to add with increasing slit width.)
3.
Hybrid emission: Light throughput increases
with the square of slit-width over a limited
wavelength range, then plateaus.
D.Know your spectrometer
A spectrometer system consists of three major sub-
assemblies:
1.
A spectrometer: We will use the PARISS
imaging spectrometer as an example.
2.
A spectrum detector: We will use a CCD or
CMOS camera as the spectrum detector.
3.
Fore-optics: Elements that precede the
spectrometer that collect light from the FOV and
delivers it to the spectrometer. Could be
microscope objectives, telescopes, focusing
mirrors, lenses.
E. Assembly details
1.
Imaging spectrograph: A spectrometer that
delivers spatial resolution along the length of the
slit, perpendicular to the wavelength dispersion
axis, and along the dispersion axis itself.
2.
A spectrograph becomes a spectrometer when
equipped with a spectral output device such as
a camera.
3.
Wavelength dispersion (WD): measured in
nm/mm. WD is more or less constant with a
diffraction grating, but varies non-linearly with a
prism. Note well: non-linear wavelength
dispersion significantly extends the useful
wavelength range of a spectrometer compared
to a diffraction grating. It is not a disadvantage.
See here
4.
Entrance slit (ES): The ES is characterized by
its slit-width (w) and slit height (h). The slit-width
controls the bandpass and resolution of the
instrument. As such, it is one of the most critical
parameters of the instrument.
All spectrometers image the ES onto an exit
plane (hence the origin of the term “line spectra”
when characterizing elemental ion-type
emissions).
The width of an individual line feature is the
projected width of ES, and determines the
spectral bandpass as a function of WD. The
height of ES determines the spatial component.
5.
Bandpass (BP): BP = Slit-width x wavelength
dispersion (BP= w x WD) or the width of camera
pixels, whichever is greater.
There can be a conflict between the need for a
certain spectral resolution (bandpass) and the
optimum slit-width needed for maximum light
throughput.
6.
Bandpass (FWHM): is the observed Full Width
at Half Maximum (FWHM) of an “infinitely”
narrow line spectrum at the exit plane of the
spectrometer.
(In practice, use a low-pressure Hg lamp, a
MIDL lamp, or a single-mode laser to check the
actual bandpass.)
The FWHM should be three pixels wide for an
optimized camera or as many pixels that
correspond to the width of the image of the
entrance slit.
If the width of camera pixels is larger than the
image of the entrance slit, then the camera
pixels determine spectral bandpass.
7.
Bandpass or resolution? Resolution is the
bandpass (w x WD) when reducing w results in
no reduction in bandpass.
“Resolution” is a function of residual aberrations
and geometric optics (wavelength, Airy disk…)
in the optics.
The terms “Bandpass” and “spectral resolution”
are only interchangeable when the entrance slit
is at its narrowest (also, light throughput will be
at its lowest - see why later in this document)
8.
Spectral range on the camera chip: The
wavelength dispersion determines the length of
a spectral range in mm. If the wavelength range
is 365- 920-nm over 8-mm, then the chip must
be at least 8-mm in width. The height of the
chip must exceed the height of the entrance slit.
9.
Spectral distribution on the camera chip: The
spectrum detector will be a CMOS or CCD
camera. Spectral distribution is along pixel rows
(in “x”). Spatial resolution of objects imaged on
the entrance slit along columns (in “y”).
10.
Characteristics of wavelength dispersive
element
a.
Diffraction grating efficiency peaks at a
single wavelength (the blaze wavelength)
and tapers off rapidly at wavelengths shorter
and more gradually to wavelengths longer
than blaze. At best, peak efficiency will be
around <60-70% at blaze. Most light losses
are due to light diffracted into higher orders.
There can also be a polarization component
that contributes to lower efficiency.
b.
A prism refracts light: Therefore higher-order
problems do not exist. A prism offers
inherent light transmission in above 90%
from 400-1000-nm and greater than <80% at
365-nm. There is no intrinsic polarization
factor.
A prism is ideal when a light source presents
weak spectra.
11.
Impact of entrance slit-width: The width of the
entrance slit should be at least three pixels
wide. In practice, the PARISS spectrograph is
supplied with a fixed entrance slit-width, either
25, 50 or 100-micron.
12.
Impact of spectrometer focal length: Wavelength
dispersion varies linearly with focal length. Long
focal length instruments deliver higher resolution
than short focal length instruments and can
accommodate wider entrance slit widths.
13.
Pixel-size considerations: Many spectroscopic
applications present objects in the FOV that
vary considerably in brightness.
a.
Small pixel camera chip (<5-micron): Small
pixels lack the electron well-depth required
to handle the linear dynamic range of weak
and strong signals from the FOV. A good
partial solution, in this case, is to bin pixels,
thereby creating a “larger” pixel with an
increase in well-depth. The increase will not
be linear, but is still an improvement over
small, non-binned pixels.
b.
Large pixel, greater than 15-micron camera
chip: The image of a 25-micron entrance slit
will under-fill three pixels resulting in a loss
of bandpass and efficiency. In this case, it is
best to increase the slit-width to 50-micron.
F.Light throughput is all about etendue (also
called “geometric extent.”)
Figure 1: Geometrical optic setup
Definitions
O: Objects in the field-of-view (FOV) with area S
S:Area of objects in the FOV
L1: Light collection fore-optics can be a lens or
front-surface mirror.
L1 is responsible for bringing light from the FOV to
the spectrometer.
Ω1:Entrance half-angle subtended by light collection
optic L1
p: Entrance path distance from FOV to L1
q: Exit path distance from L1 to the entrance slit
Ω2: Exit half-angle subtended by L1 to image plane
at the entrance slit
L1S: Illuminated area of L1
ES:Entrance slit of the spectrometer
S2: Area of the entrance slit. Should accommodate
the image of objects in the FOV.
w: Entrance slit-width (3 or more camera pixels
wide)
h:Entrance slit-height
O1: Magnified image of objects (q/p) in the FOV with
area S2
SWD: Area of the prism or grating wavelength
dispersive element: (PARISS prism shown)
B: Radiance (Luminance) The intensity when spread
over a given surface.
Intensity/Surface area of the source in
watts/steradian/cm2.
B varies inversely with the square of either p or q
Geometric etendue (geometric extent), G,
characterizes the ability of an optical system to
accept light. It is a function of the area, S, of the
emitting source and the solid angle, Ω, into which it
propagates. See figure 1, then:
G in terms of bandpass:
G can be simplified to:
and so on, for all optics in the train. All these
parameters are either known or easily determined
Magnification =
And brightness = B
G. Summary takeaways
1.
Identify the elements in the optical train that limit
etendue, and you identify the light throughput
bottle-neck.
2.
For narrow bandpass emissions, throughput
varies linearly with the area of the entrance slit.
Therefore, given that spectral bandpass varies
linearly with slit-width, go for the maximum slit-
width possible for a required bandpass
3.
For continuum spectra, throughput varies as the
square of the slit width
4.
BP varies linearly with slit-width
5.
Throughput varies with the square of the ratio of
f-number or numerical aperture (conditions
familiar to most, if not all, photographers.) (1.4)
6.
Brightness B varies with the square of the
magnification (1.8)
7.
The slit-width should correspond to the size of
the image of objects in the FOV.
8.
All cones of light described by NA or f-number
must result in perfectly filled optics or apertures.
(1.4)
9.
Objects in the distance require collection optics
with a large collection area, such as a
telescope.
10.
Near-field objects require high NA optics, such
as a microscope objective
H. Worked example optimizing f/2 collection to
work with an f/5 spectrometer:
Simplistically, for a thin lens
Where
F = focal length (Focus of an object at infinity)
p = distance of lens to FOV (object distance)
q = distance of lens to spectrometer (Image
distance)
D = diameter
Then selecting the nearest online catalog achromat
(*Edmund Optics #32-319)
D = 25-mm
F = 35-mm
p = 49-mm
q = 125-mm
Od = Object diameter = up to 9-micron (e.g.
biological cells)
Then:
1.
Spectrometer: f/5
2.
Entrance optic EO diameter D = 25-mm
aperture masked to 24.5-mm
3.
Distance from EO to the FOV = q = 49
4.
Therefore, the entry f-number of the EO = p/d =
49/24.5 = 2
5.
Distance from EO to the entrance slit = p = 125
6.
The exit f-number = 125/24.5 = f/5.1 (Under-
fills the spectrometer so all good!)
7.
Magnification = q/p = 125/49 = 2.6
8.
The image of the object = 9*2.6 = 23.4-micron
and is contained within ES.
9.
Brightness will decrease by a factor of 0.15x
(1.8)
All photons in the original object are distributed over
a greater area thereby reducing brightness by a
factor of 0.15x. However, NO photons were lost .
I.
Optimizing a light light collection of distant
objects to work with an f/5 spectrometer:
Selecting the nearest online catalog achromat
(*Edmund Optics #32-327)
D = 25-mm
F = 100-mm
q = 125-mm(distance of L1 to the spectrometer)
p = 500-mm (distance of L1 to the FOV)
Od = Object diameter = up to 60-micron
Then:
1.
The effective exit f-number of the EO = q/d =
125/25 = 5 = the f-number of the spectrometer
(as in the previous example)
2.
De-magnification = 125/500 = 0.4x
3.
The image of a 60-micron objects is de-
magnified by 4 resulting in 24-micron images.
(60*0.4 = 24-micron)
4.
For a 25-micron wide standard entrance slit-
width = each 24-micron image of the object will
be captured by the entrance slit, without light
loss.
5.
The brightness of the image will increase by a
factor of 16x.
6.
The light collection of the system will be f/20
In each of the above examples, the entrance slit fully
accommodates the projected image of the objects in
the FOV and the illuminating light cones of light
match the spectrometer f-number.
J.Summary, limitations, caveats and constraints
1.
Off-the-shelf parts entrance optics are unlikely
to fulfill all the parameters required for optimum
system performance. The stock lenses shown in
the examples are for illustration purposes only,
and are NOT recommended or endorsed.
2.
The area occupied by the wavelength range of
the spectrometer is almost never negotiable. If
the minimum and maximum wavelengths in the
range occupy a given distance, the camera chip
must be large enough to accommodate it.
3.
Imaging the entrance slit: All wavelength
dispersive spectrometers image the entrance slit
onto the spectrum detector. However, to meet
the Rayleigh and Nyquist criteria for optimum
resolution and light throughput, the slit-width
should be at least three pixels wide. Choosing
the optimum slit-width should be discussed with
the spectrometer manufacturer.
4.
Camera pixel size: Two points of consideration
a.
small pixels limit well-depth and linear
dynamic range
b.
Large pixels limit spectral bandpass due to
the increased slit-width needed to
accommodate three pixels.
5.
Light collection entry optics: Working with
commercial photographic camera lenses can be
a challenge when optimizing a spectrometer
system.
You will need to obtain data sheets that detail
chromatic aberration over your wavelength
range, coatings that cut off at certain
wavelengths, and interfacing constraints.
Most spectrometers, including PARISS, use c-
mount lenses; however, commercial
photographic camera lenses use a variety of
interfaces, including bayonet fittings. Adapters
are available to convert to c-mount, but overall
performance often degrades.