Section 5: The
Relationship Between Wavelength and Pixel Position on an Array
For a monochromator system
being used in spectrograph configuration with a solid state detector
array, the user should be aware of the following
(a) The focal plane may be
tilted by an angle, gamma. Therefore, the pixel position normally
occupied by the exit slit may NOT mark the normal to the focal plane.
(b) The dispersion and
image magnification may vary over the focal plane.
(c) As a consequence of
(b), the number of pixels per bandpass may vary not only across the
focal plane but will also vary depending on the wavelength coverage.
Figure 21(a) illustrates a
tilted focal plane that may be present in Czerny - Turner
monochromators. In the case of aberration corrected holographic
gratings, gamma, betaH.
and LH
are provided as standard operating parameters.
Operating manuals for many
Czerny Turner (CZ) and Fastie Ebert (FE) monochromators
rarely provide information on the tilt of the focal plane, therefore,
it may be necessary for the user to deduce the value of gamma. This
is most easily achieved by taking a well known spectrum and
iteratively substituting incremental values of +/- gamma, until the
wavelength appearing at each pixel corresponds to calculated values.
Figure 21 -
Spectrograph with Focal Plane (a) inclined and (b) Normal to the
Central Wavelength
5.1 The Determination of Wavelength at a Given Location on a Focal Plane
The terms used below are
consistent for aberration corrected holographic concave
gratings as well as Czerny Turner and Fastie Ebert spectrometers.
Lambda C
- Wavelength (in nm) at center of array (where exit slit would
usually be located)
LA
- Entrance arm length (mm)
L B
lambda n -
Exit arm length to each wavelength located on the focal plane (mm)
L B
lambda c -
Exit arm length to lambda c
(Czerny Turner and Fastie Ebert monochromators LA
= L B lambda c =F)
LH
- Perpendicular distance from grating or focusing mirror to the focal
plane (mm)
F - Instrument "focal
length". For CZ and FE monochromators LA
= F = LB.
(mm)
beta H
- Angle from LH
to the normal to the grating (this will vary in a scanning instrument)
beta lambda
n - Angle of
diffraction at wavelength n
beta lambda
c - Angle of
diffraction at center wavelength
H B
lambda n -
Distance from the intercept of the normal to the focal plane to the
wavelength lambda n
H B
lambda c -
Distance from the intercept of the normal to the focal plane to the
wavelength lambda c
Pmin
- Pixel # at extremity corresponding to lambda min
(e.g., # 1)
Pmax
- Pixel # at extremity corresponding to lambda max
(e.g., # 1024)
Pw
- Pixel width (mm)
Pc
- Pixel # at lambda c
(e.g., # 512)
P lambda
- Pixel # at lambda n
Inclination of the focal
plane measured at the location normally occupied by the exit slit,
lambda c.
(This is usually the center of the array. However, provided that the
pixel marking this location is known, the array may be placed as the
user finds most useful). For this reason, it is very convenient to
use a spectrometer that permits simple interchange from scanning to
spectrograph by means of a swing away mirror. The instrument
may then be set up with a standard slit using, for example, a mercury
lamp. Switching to spectrograph mode enables identification of the
pixel, Pc,
illuminated by the wavelength previously at the exit slit.
The equations that follow
are for Czerny Turner type instruments where gamma = 0° in
one case and gamma does not equal 0 in the other.
Case 1 gamma = 0°.
See Figure 21(b).
LH
= LB
= F at lambda c (mm)
beta H
= beta at lambda c
HB lambda
n = Pw
(P lambda
- Pc)
(mm)
HB is negative for
wavelengths shorter than lambda c.
HB is positive for
wavelengths longer than lambda c.
beta lambda
n = betaH
tan-1
(HB lambda n /LH)
(5l)
Note: The secret of
success (and reason for failure) is frequently the level of
understanding of the sign convention. Be consistent, make reasonably
accurate sketches whenever possible and be philosophical about the
arbitrary nature of the beast.
To make a calculation,
alpha and beta at lambda c
can be determined from Equations (12)
and (21).
At this point the value for alpha is used in the calculation of all
values beta lambda
n for each wavelength.
Then
 (52)
Case 2: gamma does not
equal 0°
See Figure 21(a).
LH
= F cos gamma (where F = L B
lambda c) (5-3)
betaH
= beta lambda c
+ gamma (54)
HB lambda
c = F sin
gamma (55)
HB lambda
n = Pw
(P lambda
Pc)
+ HB lambda c (56)
beta lambda
n = betaH
tan-1
(HB lambda n /LH)
(57)
Again keeping significant
concern for the sign of HB lambda
n, proceed
to calculate the value beta lambda
n after
first obtaining alpha at lambda c
Then use Equation (52) to calculate lambda n.
IN PRACTICE, THIRD AND
FOURTH DECIMAL PLACE ACCURACY IS NECESSARY.
Indeed the longer the
instrument's focal length, the greater the contribution of rounding errors.
To illustrate the above
discussion a worked example, taken from a readily available
commercial instrument, is provided.
Example:
The following are typical
results for a focal plane inclined by 2.4° in Czerny
Turner monochromator used in spectrograph mode.
LB
= 320 mm at lambda c
= F
n = 1800 g/mm
D = 24°
LH
= 319.719 mm
gamma = 2.4°
HB lambda
c = 13.4 mm
Array length = 25.4 mm;
lambda c
appears 12.7 mm from end of array
lambda min,
lambda max
= wavelength at array extremities
lambda error
min, max =
wavelength thought to be at array extremity if gamma = 0°
Disp = dispersion
(Equation (15))
(nm/mm)
mag = magnification in
dispersion plane (Equation (216))
delta lambda (gamma =
0°) lambda min
or lambda max
- lambda error (nm)
delta d = Actual distance
of lambda error
from extreme
pixel (um)
Table 7 Operating
Parameters for a CZ Spectrometer with a 2.4° Tilt at lambda c
on the Spectral Plane Compared to a 0° Tilt.
|
nm |
lambda min 229.9463 |
lambda c 250 |
lambda max
269.7469 |
lambda min 381.4545 |
lambda c 400 |
lambda max
418.1236 |
lambda min 686.1566 |
lambda c 700 |
lambda max
713.1999 |
|
alpha |
1.29864 |
9.5950 |
28.0963 |
|
beta H |
27.6986 |
35.9950 |
54.496 |
|
beta |
23.0317 |
25.2986 |
27.5732 |
31.3280 |
33.5950 |
35.8695 |
49.8294 |
52.0963 |
54.3707 |
|
Disp. |
1.59 |
1.57 |
1.54 |
1.48 |
1.45 |
1.41 |
1.12 |
1.07 |
1.01 |
|
Mag |
1.09 |
1.11 |
1.13 |
1.16 |
1.18 |
1.22 |
1.37 |
1.44 |
1.51 |
|
delta lambda |
0.051 |
0 |
0.015 |
0.048 |
0 |
0.014 |
0.037 |
0 |
0.011 |
|
delta d |
+32 |
0 |
-10 |
+32 |
0 |
-10 |
+32 |
0 |
-10 |
5.1.1 Discussion of Results
Examination of the results
given in the worked example indicates the following phenomena:
A. If an array with 25 um
pixels was used and the focal plane was assumed to be normal to
lambda c
rather than the actual 2.4°, at least a one pixel error (32 um)
would be present at lambda min.
(This may not seem like much, but it is incredible how much lost
sleep and discussion time has been spent attempting to rationalize
this dilemma).
B. A 25 um entrance slit
is imaged in the focal plane with a width of 27.25 um (1.09 x 25) at
229.946 nm (when lambda c
= 250 nm) but is imaged with a width of 37.75 um at 713.2 nm (1.51 x
25) (when lambda c
= 700 nm), Indeed in this last case the difference in image width at
lambda min
compared to lambda max
varies by over 10% across the array.
C. If the array did not
limit the resolution, then a 25 um entrance slit width would produce
a bandpass of 0.04 nm. Given that, in the above example with gamma =
0° rather than 2.4°, the wavelength error at lambda min
exceeds 0.04 nm. Therefore, a spectral line at this extreme end of
the spectral field could "disappear" the closer lambda c
comes to the location of the exit slit.
D. The spectral coverage
over the 25.4 mm array varies in the examples calculated as follows:
|
|
|
lambda c (nm) |
(lambda max - lambda min) (nm) |
|
250 |
39.80 |
|
400 |
36.67 |
|
700 |
27.04 |
5.1.2 Determination of the Position of a known Wavelength In the
Focal Plane
In this case, provided
lambda c
is known, alpha, betaH,
and LH
may be determined as above. If lambda n
is known, the beta lambda
n may be
obtained from the Grating Equation (11). Then
HB lambda
n = LH
tan (betaH
- beta lambda n)
(58)
This formula is most
useful for constructing alignment targets with the location of known
spectral lines marked on a screen or etched into a ribbon, etc.
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