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Section 2:Monochromators & Spectrographs
2.1
Basic Designs
Monochromator and
spectrograph systems form an image of the entrance slit in the exit
plane at the wavelengths present in the light source. There are
numerous configurations by which this may be achieved -- only the
most common are discussed in this document and includes Plane Grating
Systems (PGS) and Aberration Corrected Holographic Grating (ACHG) systems.
Definitions
LA
- entrance arm length
LB
- exit arm length
h - height of entrance slit
h' - height of image of
the entrance slit
alpha - angle of incidence
beta - angle of diffraction
w - width of entrance slit
w' - width of entrance
slit image
Dg - diameter of a
circular grating
Wg - width of a
rectangular grating
Hg - height of a
rectangular grating
2.2
Fastie-Ebert Configuration
A Fastie-Ebert instrument
consists of one large spherical mirror and one plane diffraction
grating (see
Fig. 6).
A portion of the mirror
first collimates the light which will fall upon the plane grating. A
separate portion of the mirror then focuses the dispersed light from
the grating into images of the entrance slit in the exit plane.
It is an inexpensive and
commonly used design, but exhibits limited ability to maintain image
quality offaxis due to system aberrations such as spherical
aberration, coma, astigmatism, and a curved focal field.
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2.3
Czerny-Turner Configuration
The Czerny-Turner (CZ)
monochromator consists of two concave mirrors and one plano
diffraction grating (see
Fig. 7).
Although the two mirrors
function in the same separate capacities as the single spherical
mirror of. the Fastie-Ebert configuration, i.e., first collimating
the light source (mirror 1), and second, focusing the dispersed light
from the grating (mirror 2), the geometry of the mirrors in the
Czerny-Turner configuration is flexible.
By using an asymmetrical
geometry, a Czerny-Turner configuration may be designed to produce a
flattened spectral field and good coma correction at one wavelength.
Spherical aberration and astigmatism will remain at all wavelengths.
It is also possible to
design a system that may accommodate very large optics. |
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Figure 7 - Czerny-Turner Configuration |
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2.4
Czerny-Turner/Fastie-Ebert PGS Aberrations
PGS spectrometers exhibit
certain aberrations that degrade spectral resolution, spatial
resolution, or signaltonoise ratio. The most significant
are astigmatism, coma, spherical aberration and defocusing. PGS
systems are used offaxis, so the aberrations will be different
in each plane. It is not within the scope of this document to review
the concepts and details of these aberrations, (reference 4) however,
it is useful to understand the concept of Optical Path Difference
(OPD) when considering the effects of aberrations.
Basically, an OPD is the
difference between an actual wavefront produced and a "reference
wavefront that would be obtained if there were no aberrations. This
reference wavefront is a sphere cantered at the image or a plane if
the image is at infinity. For example:
1) Defocusing results in
rays finding a focus outside the detector surface producing a blurred
image that will degrade bandpass, spatial resolution, and optical
signal-to-noise ratio. A good example could be the spherical
wavefront illuminating mirror M1 in Fig. 7. Defocusing should not be
a problem in a PGS monochromator used with a single exit slit and a
PMT detector. However, in an uncorrected PGS there is field curvature
that would display defocusing towards the ends of a planar linear
diode array. Geometrically corrected CZ configurations such as that
shown in Fig. 7 nearly eliminate the problem. The OPD due to
defocusing varies as the square of the numerical aperture.
2) Coma is the result of
the off-axis geometry of a PGS and is seen as a skewing of rays in
the dispersion plane enlarging the base on one side of a spectral
line as shown in Fig. 8. Coma may be responsible for both degraded
bandpass and optical signal-to-noise ratio. The OPD due to coma
varies as the cube of the numerical aperture. Coma may be corrected
at one wavelength in a CZ by calculating an appropriate operating
geometry as shown in Fig. 7.
3) Spherical aberration is
the result of rays emanating away from the centre of an optical
surface failing to find the same focal point as those from the centre (See
Fig. 9). The
OPD due to spherical aberration varies with the fourth power of the
numerical aperture and cannot be corrected without the use of
aspheric optics.
4) Astigmatism is
characteristic of an off-axis geometry. In this case a spherical
mirror illuminated by a plane wave incident at an angle to the normal
(such as mirror M2 in Fig.
7) will
present two foci: the tangential focus, Ft,
and the sagittal focus, FS.
Astigmatism has the effect of taking a point at the entrance slit
and imaging it as a line perpendicular to the dispersion plane at the
exit (see
Fig. 10),
thereby preventing spatial resolution and increasing slit height with
subsequent degradation of optical signaltonoise ratio. The
OPD due to astigmatism varies with the square of numerical aperture
and the square of the offaxis angle and cannot be corrected
without employing aspheric optics.
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2.4.1
Aberration Correcting Plane Gratings
Recent advances in
holographic grating technology now permits complete correction of ALL
aberrations present in a spherical mirror based CZ spectrometer at
one wavelength with excellent mitigation over a wide wavelength range
(Ref. 12).
2.5
Concave Aberration Corrected Holographic Gratings
Both the monochromators
and spectrographs of this type use a single holographic grating with
no ancillary optics.
In these systems the
grating both focuses and diffracts the incident light.
With only one optic in
their design, these devices are inexpensive and compact. Figure 11a
illustrates an ACHG monochromator. Figure 11b illustrates an ACHG
spectrograph in which the location of the focal plane is established by:
betaH
- Angle between perpendicular to spectral plane and grating normal.
LH
- Perpendicular distance from spectral plane to grating.
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2.6
Calculating alpha and beta in a Monochromator Configuration
From Equation (1-2),
 (remains
constant)
Taking this equation and
Equation (1-3),
 (21)
Use Equations (21)
and (12) to determine alpha and beta, respectively. See Table 3
for worked examples.
Note: In practice the
highest wavelength attainable is limited by the mechanical rotation
of the grating. This means that doubling the groove density of the
grating will halve the spectral range. (See
Section 2.14).
2.7
Monochromator System Optics
To understand how a
complete monochromator system is characterized, it is necessary to
start at the transfer optics that brings light from the source to
illuminate the entrance slit. . (See
Fig. 12)Here
we have "unrolled" the system and drawn it in a linear fashion.
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AS - aperture stop
L1 - lens 1
M1 - mirror 1
M2 - mirror 2
G1 - grating
p - object distance to
lens L1
q - image distance from
lens L1
F - focal length of lens
L1 (focus of an object at infinity)
d - the clear aperture of
the lens (L1 in diagram)
omega - half-angle
s - area of the source
s' - area of the image of
the source
2.8
Aperture Stops and Entrance and Exit Pupils
An aperture stop (AS)
limits the opening through which a cone of light may pass and is
usually located adjacent to an active optic.
A pupil is either an
aperture stop or the image of an aperture stop.
The entrance pupil of the
entrance (transfer) optics in Fig. 12 is the virtual image of AS as
seen axially through lens L1 from the source.
The entrance pupil of the
spectrometer is the image of the grating (G1) seen axially through
mirror M1 from the entrance slit.
The exit pupil of the entrance
optics is AS
itself seen axially from the entrance slit of the spectrometer.
The exit pupil of the spectrometer
is the image of the grating seen axially through M2 from the exit slit.
2.9
Aperture Ratio (f/value, F.Number), and Numerical Aperture (NA)
The light gathering power
of an optic is rigorously characterized by Numerical Aperture(NA).
Numerical Aperture is
expressed by:
 (22)
and f/value by:
 (23)
|
Table2: Relationship between f/value,
half-angle, and numerical aperture |
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f/value |
f/2 |
f/3 |
f/5 |
f/7 |
f/10 |
f/15 |
|
n (degrees) |
14.48 |
9.6 |
5.7 |
4.0 |
2.9 |
1.9 |
|
NA |
0.25 |
0.16 |
0.10 |
0.07 |
0.05 |
0.03 |
2.9.1
f/value of a Lens System
f/value is also given by
the ratio of either the image or object distance to the diameter of
the pupil. When, for example, a lens is working with finite
conjugates such as in Fig. 12, there is an effective f/value from the
source to L1 (with diameter AS) given by:
effective
f/valuein=
(P/diameter of entrance pupil) = (P/image of AS) (24)
and from L1 to the
entrance slit by:
effective
f/valueout=
(q/diameter of exit pupil) = (q/AS) (25)
In the sections that
follow f/value will always be calculated assuming that the entrance
or exit pupils are equivalent to the aperture stop for the lens or
grating and the distances are measured to the center of the lens or grating.
When the f/value is
calculated in this way for f/2 or greater (e.g. f/3, f/4, etc.), then
sin omega is ~ tan omega and the approximation is good. However, if
an active optic is to function at an f/value significantly less than
f/2, then the f/value should be determined by first calculating
Numerical Aperture from the half-angle.
2.9.2
f/value of a Spectrometer
Because the angle of
incidence alpha is always different in either sign or value from the
angle of diffraction beta (except in Littrow), the projected size of
the grating varies with the wavelength and is different depending on
whether it is viewed from the entrance or exit slits. In Figures 13a
and 13b, the widths W' and W'' are the projections of the grating
width as perceived at the entrance and exit slits, respectively.
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To determine the f/value
of a spectrometer with a rectangular grating, it is first necessary
to calculate the "equivalent diameter", D', as seen from
the entrance slit and D" as seen from the exit slit. This is
achieved by equating the projected area of the grating to that of a
circular disc and then calculating the diameter D' or D".
W'g = Wg cos alpha =
projected area of grating from entrance slit (26)
W"g = Wg cos beta =
projected area of grating from exit slit (27)
In a spectrometer,
therefore, the f/valuein
will not equal the f/valueout.
f/valuein
= LA/D'(28)
f/valueout
= LB/D"(29)
where, for a rectangular
grating, D' and D" are given by:
 (210)
 (211)
where, for a circular
grating, D' and D" are given by:
D' = Dg(cos
alpha)^(1/2) (212)
D" = Dg(cos
beta)^(1/2) (213)
Table 3 shows how the
f/value changes with wavelength.
Table 3 Calculated values
for f/valuein
and f/valueout
for a Czerny-Turner configuration with 68 x 68 mm, 1800 g/mm grating
and LA
= LB
= F = 320 nm. Dv = 24 °.
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Lambda(nm) |
alpha |
beta |
f/valuein |
f/valueout |
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200 |
1.40 |
22.60 |
4.17 |
4.34 |
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320 |
5.12 |
29.12 |
4.18 |
4.46 |
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500 |
15.39 |
39.39 |
4.25 |
4.74 |
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680 |
26.73 |
50.73 |
4.41 |
5.24 |
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800 |
35.40 |
59.40 |
4.62 |
5.84 |
2.9.3
Magnification and Flux Density
In any spectrometer system
a light source should be imaged onto an entrance slit (aperture)
which is then imaged onto the exit slit and so on to the detector,
sample, etc. This process inevitably results in the magnification or
demagnification of one or more of the images of the light source.
Magnification may be determined by the following expansions, taking
as an example the source imaged by lens L1 in Fig. 12 onto the
entrance slit:
 (214)
Similarly, flux density is
determined by the area that the photons in an image occupy, so
changes in magnification are important if a flux density sensitive
detector or sample are present. Changes in the flux density in an
image may be characterized by the ratio of the area of the object, S,
to the area of the image, S', from which the following expressions
may be derived:
 (215)
These relationships show
that the area occupied by an image is determined by the ratio of the
square of the f/values. Consequently, it is the EXIT f/value that
determines the flux density in the image of an object. Those using
photographic film as a detector will recognize these relationships in
determining the exposure time necessary to obtain a certain
signal-to-noise ratio.
2.10
Exit Slit Width and Anamorphism
Anamorphic optics are
those optics that magnify (or demagnify) a source by different
factors in the vertical and horizontal planes. (See
Fig. 14).
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In the case of a
diffraction grating-based instrument, the image of the entrance slit
is NOT imaged 1:1 in the exit plane (except in Littrow and
perpendicular to the dispersion plane assuming LA
= LB).
This means that in
virtually all commercial instruments the tradition of maintaining
equal entrance and exit slit widths may not
always be appropriate.
Geometric horizontal
magnification depends on the ratio of the cosines of the angle of
incidence, alpha, and the angle of diffraction, beta, and the LB/LA
ratio (Equation (216)). Magnification may change substantially
with wavelength. (See Table 4).
 (216)
Table 4 illustrates the
relationship between alpha, beta, dispersion, horizontal
magnification of entrance slit image, and bandpass.
Table 4 Relationship
Between Dispersion, Horizontal Magnification, and Bandpass in a
CzernyTurner Monochromator. LA
= 320 mm, LB
= 320 mm, Dv = 24 deg, n = 1800 g/mm Entrance slit width = 1 mm
|
Wavelength (nm) |
alpha (deg.) |
beta (deg.) |
dispersion (nm/mm) |
horiz. magnif. |
bandpass* (nm) |
|
200 |
-1.4 |
22.60 |
1.60 |
1.08 |
1.74 |
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260 |
1.84 |
25.84 |
1.56 |
1.11 |
1.74 |
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320 |
5.12 |
29.12 |
1.46 |
1.14 |
1.73 |
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380 |
8.47 |
32.47 |
1.41 |
1.17 |
1.72 |
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440 |
11.88 |
35.88 |
1.34 |
1.21 |
1.70 |
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500 |
15.39 |
39.39 |
1.27 |
1.25 |
1.67 |
|
560 |
19.01 |
43.01 |
1.19 |
1.29 |
1.64 |
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620 |
22.78 |
46.78 |
1.10 |
1.35 |
1.60 |
|
680 |
26.73 |
50.73 |
1.00 |
1.41 |
1.55 |
|
740 |
30.91 |
54.91 |
0.88 |
1.49 |
1.49 |
|
800 |
35.40 |
59.40 |
1.60 |
1.60 |
1.42 |
Exit slit width matched to
image of entrance slit.
*As the inclination of the
grating becomes increasingly large, coma in the system will increase.
Consequently, in spite of the fact that the bandpass at 800 nm is
superior to that at 200 nm, it is unlikely that the full improvement
will be seen by the user in systems of less than f/8.
2.11
Slit Height Magnification
Slit height magnification
is directly proportional to the ratio of the entrance and exit arm
lengths and remains constant with wavelength (exclusive of the
effects of aberrations that may be present).
h' = (LB/LA)h
(217)
Note: Geometric
magnification is not an aberration!
2.12
Bandpass and Resolution
In the most fundamental
sense both bandpass and resolution are used as measure of an
instrument's ability to separate adjacent spectral lines.
Assuming a continuum light
source, the bandpass (BP) of an instrument is the spectral interval
that may be isolated. This depends on many factors including the
width of the grating, system aberrations, spatial resolution of the
detector, and entrance and exit slit widths.
If a light source emits a
spectrum which consists of a single monochromatic wavelength lambdao (Fig.
15) and is
analyzed by a perfect spectrometer, the output should be identical to
the spectrum of the emission (Fig.
16) which is a
perfect line at precisely lambdao. |
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In reality spectrometers
are not perfect and produce an apparent spectral broadening of the
purely monochromatic wavelength. The line profile now has finite
width and is known as the "instrumental line profile"
(instrumental bandpass). (See
Fig. 17).
The instrumental profile
may be determined in a fixed grating spectrograph configuration with
the use of a reasonably monochromatic light source such as a single
mode dye laser. For a given set of entrance and exit slit parameters,
the grating is fixed at the proper orientation for the central
wavelength of interest and the laser light source is scanned in
wavelength. The output of the detector is recorded and displayed. The
resultant trace will show intensity versus wavelength distribution.
For a monochromator the
same result would be achieved if a monochromatic light source is
introduced into the system and the grating rotated.
The bandpass is then
defined as the Full Width at Half Maximum (FWHM) of the trace
assuming monochromatic light.
Any spectral structure may
be considered to be the sum of an infinity of single monochromatic
lines at different wavelengths. Thus, there is a relationship between
the instrumental line profile, the real spectrum and the recorded spectrum.
Let B(lambda) be the real
spectrum of the source to be analysed.
Let F(lambda) be the
recorded spectrum through the spectrometer.
Let P(lambda) be the
instrumental line profile.
F = B * P (218)
The recorded function
F(lambda) is the convolution of the real spectrum and the
instrumental line profile.
The shape of the
instrumental line profile is a function of various parameters:
Each of these factors may
be characterized by a special function Pi(lambda), each obtained by
neglecting the other parameters. The overall instrumental line
profile P(lambda) is related to the convolution of the individual terms:
P(lambda) = P1(lambda) *
P2(lambda) *.....* Pn(lambda) (219)
2.12.1
Influence of the Slits (P1(lambda))
If the slits are of finite
width and there are no other contributing effects to broaden the
line, and if:
Went
= width of the image of the entrance slit
Wex
= width of the exit slit or of one pixel in the case of a
multichannel detector
delta lambda 1
= linear dispersion x Went
delta lambda 2
= linear dispersion x Wex
then the slit's
contribution to the instrumental line profile is the convolution of
the two slit functions. (See
Fig. 18).
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2.12.2
Influence of Diffraction (P2(lambda))
If the two slits are
infinitely narrow and aberrations negligible, then the instrumental
line profile is that of a classic diffraction pattern. In this case,
the resolution of the system is the wavelength, lambda, divided by
the theoretical resolving power of the grating, R (Equation
111).
2.12.3
Influence of Aberrations (P3(lambda))
If the two slits are
infinitely narrow and broadening of the line due to aberrations is
large compared to the size due to diffraction, then the instrumental
line profile due to diffraction is enlarged.
2.12.4
Determination of the FWHM of the Instrumental Profile
In practice the FWHM of
F(lambda) is determined by the convolution of the various causes of
line broadening including:
d lambda (resolution): the
limiting resolution of the spectrometer is governed by the limiting
instrumental line profile and includes system aberrations and
diffraction effects.
d lambda (slits): bandpass
determined by finite spectrometer slit widths.
d lambda (line): natural
line width of the spectral line used to measure the FWHM.
Assuming a gaussian line
profile (which is not the case), a reasonable approximation of the
FWHM is provided by the relationship:
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 (220) |
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In general, most
spectrometers are not routinely used at the limit of their resolution
so the influence of the slits may dominate the line profile. From
Fig. 18 the FWHM, due to the slits, is determined by either the image
of the entrance slit or the exit slit, whichever is greater. If the
two slits are perfectly matched and aberrations minimal compared to
the effect of the slits, then the FWHM will be half the width at the
base of the peak. (Aberrations may, however, still produce broadening
of the base). Bandpass (BP) is then given by:
BP = FWHM ~ linear
dispersion x (exit slit width or the image of the entrance slit,
whichever is greater).
In Section 2-10 image
enlargement through the spectrometer was reviewed. The impact on the
determination of the system bandpass may be determined by taking
Equation (216) to calculate the width of the image of the
entrance slit and multiplying it by the dispersion (Equation
(15)).
Bandpass is then given by:
 (221)
The major benefit of
optimising the exit slit width is to obtain maximum THROUGHPUT
without loss of bandpass.
It is interesting to note
from Equations (221) and (15) that:
2.12.5
Image Width and Array Detectors
Because the image in the
exit plane changes in width as a function of wavelength, the user of
an array type detector must be aware of the number of pixels per
bandpass that are illuminated. It is normal to allocate 3-6 pixels to
determine one bandpass. If the image increases in size by a factor of
1.5, then clearly photons contained within that bandpass would have
to be collected over 4-9 pixels. For a discussion of the relation
between wavelength and pixel position see Section 5. The FWHM that
determines bandpass is equivalent to the width of the image of the
entrance slit containing a typical maximum of 80% of available
photons at the wavelength of interest; the remainder is spread out in
the base of the peak. Any image magnification, therefore, equally
enlarges the base spreading the entire peak over additional pixels.
2.12.6
Discussion
a) Bandpass with
Monochromatic Light
The infinitely narrow
natural spectral band width of monochromatic light is, by definition,
less than that of the instrumental bandpass determined by Equation
(220). (A very narrow band width is typically referred to as a
"line" because of its appearance in a spectrum).
In this case all the
photons present will be at exactly the same wavelength irrespective
of how they are spread out in the exit plane. The image of the
entrance slit, therefore, will consist exclusively of photons at the
same wavelength even though there is a finite FWHM. Consequently,
bandpass in this instance cannot be considered as a wavelength spread
around the center wavelength. If, for example, monochromatic light at
250 nm is present and the instrumental bandpass is set to produce a
FWHM of 5 nm, this does NOT mean 250 nm +/- 2.5 nm because no
wavelength other than 250 nm is present. It does mean, however, that
a spectrum traced out (wavelength vs. intensity) will produce a
"peak" with an apparent FWHM of "5 nm" due to
instrumental and NOT spectral line broadening.
b) Bandpass with
"Line" Sources of Finite Spectral Width
Emission lines with finite
natural spectral bandwidths are routinely found in almost all forms
of spectroscopy including emission, Raman, fluorescence, and absorption.
In these cases spectra may
be obtained that seem to consist of line emission (or absorption)
bands. If, however, one of these "lines" is analysed with a
very high resolution spectrometer, it would be determined that beyond
a certain bandpass no further line narrowing would take place
indicating that the natural bandwidth had been reached.
Depending on the
instrument system the natural bandwidth may or may not be greater
than the bandpass determined by Equation (220).
If the natural bandwidth
is greater than the instrumental bandpass, then the instrument will
perform as if the emission "line" is a portion of a
continuum. In this case the bandpass may indeed be viewed as a
spectral spread of +/- 0.5 BP around a center wavelength at FWHM.
Example 1:
Figure 19 shows a somewhat
contrived spectrum where the first two peaks are separated on the
recording by 32 mm. The FWHM of the first peak is the same as the
second but is less than the third. This implies that the natural
bandwidth of the third peak is greater than the bandpass of the
spectrometer and would not demonstrate spectral narrowing of its
bandwidth even if evaluated with a very high resolution spectrometer.
The first and second
peaks, however, may well possess natural bandwidths less than that
shown by the spectrometer. In these two cases, the same instrument
operating under higher bandpass conditions (narrower slits) may well
reveal either additional "lines" that had previously been
incorporated into just one band, or a simple narrowing of the
bandwidth until either the limit of the spectrometer or the limiting
natural bandpass have been reached.
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Example 2:
A researcher finds a
spectrum in a journal that would be appropriate to reproduce on an
inhouse spectrometer. The first task is to determine the
bandpass displayed by the spectrum. If this information is not given,
then it is necessary to study the spectrum itself. Assuming that the
wavelengths of the two peaks are known, then the distance between
them must be measured with a ruler as accurately as possible. If the
wavelength difference is found to be 1.25 nm and this increment is
spread over 32 mm (see Fig. 19), the recorded dispersion of the
spectrum = 1.25/32 = 0.04 nm/mm. It is now possible to determine the
bandpass by measuring the distance in mm at the Full Width at Half
Maximum height (FWHM). Let us say that this is 4 mm; the bandpass of
the instrument is then 4 mm x 0.04 nm/mm = 0.16 nm.
Also assuming that the
spectrometer described in Table 4 is to be used, then from Equation
(2-21) and the list of maximum wavelengths described in Table 6, the
following options are available to produce a bandpass of 0.16 nm:
|
Table 5: Variation of Dispersion and Slit Width to Produce 0.16 nm
Bandpass in a 320 mm Focal Length Czerny-Turner |
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|
|
Groove Density (g/mm) |
Dispersion (nm/mm) |
Entrance Slit Width (microns) |
|
300 |
9.2 |
17 |
|
600 |
4.6 |
35 |
|
1200 |
2.3 |
70 |
|
1800 |
1.5 |
107 |
|
2400 |
1.15 |
139 |
|
3600 |
0.77 |
208 |
The best choice would be
the 3600 g/mm option to provide the largest slit width possible to
permit the greatest amount of light to enter the system.
2.13
Order and Resolution
If a given wavelength is
used in higher orders, for example, from first to second order, it is
considered that because the dispersion is doubled, so also is the
limiting resolution. In a monochromator in which there are ancillary
optics such as plane or concave mirrors, lenses, etc., a linear
increase in the limiting resolution may not occur. The reasons for
this include:
Even if the full width at
half maximum is maintained, a degradation in line shape will often
occur -- the base of the peak usually broadens with consequent
degradation of the percentage of available photons in the FWHM.
2.14
Dispersion and Maximum Wavelength
The longest possible
wavelength (lambda maxl) an instrument will reach mechanically with a
grating of a given groove density is determined by the limit of
mechanical rotation of that grating. Consequently, in changing from
an original groove density, n1,
to a new groove density, n2,
the new highest wavelength (lambda max2)
will be:
 (222)
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Table 6: Variation in Maximum Wavelength with Groove Denisty in a
Typical Monochromator |
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LA = LB = F = 320 mm, DV = 24 deg.;
In this example maximum wavelength at maximum possible mechanical
rotation of a 1200 g/mm grating = 1300 nm |
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|
Groove Density (g/mm) |
Dispersion (nm/mm) |
Max Wavelength (nm) |
|
150 |
18.4 |
10400 |
|
300 |
9.2 |
5200 |
|
600 |
4.6 |
2600 |
|
1200 |
2.3 |
1300 |
|
1800 |
1.5 |
867 |
|
2400 |
1.15 |
650 |
|
3600 |
0.77 |
433 |
From Table 6 it is clear
that if a 3600 g/mm grating is required to diffract light above 433
nm, the system will not permit it. If, however, a dispersion of 0.77
nm/mm is required to produce appropriate resolution at, say, 600 nm,
a system should be acquired with 640 mm focal length (Equation
(15)). This would produce a dispersion of 0.77 nm/mm with a 2400
g/mm grating and also permit mechanical rotation up to 650 nm.
2.15
Order and Dispersion
In Example 2, Section
2.12.6, the solution to the dispersion problem could be solved by
using a 2400 g/mm grating in a 640 mm focal length system. As
dispersion varies with focal length (LB),
groove density (n), and order (k); for a fixed LB
at a given wavelength, the dispersion equation (Equation
1.5)
simplifies to:
kn = constant
Therefore, if first order
dispersion = 1.15 nm/mm with a 2400 g/mm grating the same dispersion
would be obtained with a 1200 g/mm grating in second order. Keeping
in mind that k lambda = constant for a given groove density, n, (Equation
19),
using second order with an 1800 g/mm grating to solve the last
problem would not work because to find 600 nm in second order, it
would be necessary to operate at 1200 nm in first order, when it may
be seen in Table 6 that the maximum attainable first order wavelength
is 867 nm.
However, if a dispersion
of 0.77 nm/mm is necessary in the W at 250 nm, this wavelength could
be monitored at 500 nm in first order with the 1800 g/mm grating and
obtain a second order dispersion of 0.75 nm/mm. In this case any
first order light at 500 nm would be superimposed on top of the 250
nm light (and vice-versa). Wavelength selective filters may then be
used to eliminate the unwanted radiation.
The main disadvantages of
this approach are that the grating efficiency would not be as great
as an optimised first order grating and order sorting filters are
typically inefficient. If a classically ruled grating is employed,
ghosts and stray light will increase as the square of the order.
2.16
Choosing a Monochromator/Spectrograph
Select an instrument based on:
Remember: f/value is not
always the controlling factor of throughput. For example, light may
be collected from a source at f/1 and projected onto the entrance
slit of an f/6 monochromator so that the entire image is contained
within the slit. Then the system will operate on the basis of the
photon collection in the f/l cone and not the f/6 cone of the
monochromator. See
Section 3.
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