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, g. 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, g, b_H. and L_H 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 +/- g, 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.

l_c - Wavelength (in nm) at center of array (where exit slit would usually be located)
L_A - Entrance arm length (mm)
L_Bln - Exit arm length to each wavelength located on the focal plane (mm)
L_Blc - Exit arm length to l_c (Czerny ­ Turner and Fastie ­ Ebert monochromators L_A = L_Blc = F)
L_H - Perpendicular distance from grating or focusing mirror to the focal plane (mm)
F - Instrument "focal length". For CZ and FE monochromators L_A = F = L_B. (mm)
b_H - Angle from L_H to the normal to the grating (this will vary in a scanning instrument)
b_ln - Angle of diffraction at wavelength n
b_lc - Angle of diffraction at center wavelength
H_Bln - Distance from the intercept of the normal to the focal plane to the wavelength l_n
H_Blc - Distance from the intercept of the normal to the focal plane to the wavelength l_c
P_min - Pixel # at extremity corresponding to l_min (e.g., # 1)
P_max - Pixel # at extremity corresponding to l_max (e.g., # 1024)
P_w - Pixel width (mm)
P_c - Pixel # at l_c (e.g., # 512)
P_l - Pixel # at l_n
g - Inclination of the focal plane measured at the location normally occupied by the exit slit, l_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, P_c, 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 g does not equal 0 in the other.

Case 1 g = 0°.

See Figure 21(b).
L_H = L_B = F at l_c (mm)
b_H = b at l_c
H_Bln = P_w (P_l - P_c) (mm)

H_B is negative for wavelengths shorter than l_c.

H_B is positive for wavelengths longer than l_c.

b_ln = b_H ­ tan^-1 (H_Bln /L_H) (5-1)

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, a and b at l_c can be determined from Equations (1-2) and (2-1). At this point the value for a is used in the calculation of all values b_ln for each wavelength.

Then

(5-2)

Case 2: g does not equal 0°

See Figure 21(a).
L_H = F cos g (where F = L_Blc) (5-3)
b_H = b_lc + g (5-4)
H_Blc = F sin g (5-5)
H_Bln = P_w (P_l ­ P_c) + H_Blc (5-6)
b_ln = b_H ­ tan^-1 (H_Bln /L_H) (5-7)

Again keeping significant concern for the sign of H_Bln, proceed to calculate the value b_ln after first obtaining a at l_c then use Equation (5-2) to calculate l_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.

L_B = 320 mm at l_c = F
n = 1800 g/mm
D = 24°
L_H = 319.719 mm
g = 2.4°
H_Blc = 13.4 mm


Array length = 25.4 mm; l_c appears 12.7 mm from end of array
l_min, l_max = wavelength at array extremities
l_{error min}, _max = wavelength thought to be at array extremity if g = 0°
Disp = dispersion (Equation (1-5)) (nm/mm)
mag = magnification in dispersion plane (Equation (2-16))
dl(g = 0°) l_min or l_max - l_error (nm)
dd = Actual distance of l_error from extreme pixel (um)

Table 7 Operating Parameters for a CZ Spectrometer with a 2.4° Tilt at l_c on the Spectral Plane Compared to a 0° Tilt.

nm	lmin 229.9463	lc 250	lmax 269.7469	lmin 381.4545	lc 400	lmax 418.1236	lmin 686.1566	lc 700	lmax 713.1999
a	1.29864			9.5950			28.0963
bH	27.6986			35.9950			54.496
b	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
dl	0.051	0	0.015	0.048	0	0.014	0.037	0	0.011
dd	+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 mm pixels was used and the focal plane was assumed to be normal to l_c rather than the actual 2.4°, at least a one pixel error (32 mm) would be present at l_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 mm entrance slit is imaged in the focal plane with a width of 27.25 mm (1.09 x 25) at 229.946 nm (when l_c = 250 nm) but is imaged with a width of 37.75 mm at 713.2 nm (1.51 x 25) (when l_c = 700 nm), Indeed in this last case the difference in image width at l_min compared to l_max varies by over 10% across the array.

C. If the array did not limit the resolution, then a 25 mm entrance slit width would produce a bandpass of 0.04 nm. Given that, in the above example with g = 0° rather than 2.4°, the wavelength error at l_min exceeds 0.04 nm. Therefore, a spectral line at this extreme end of the spectral field could "disappear" the closer l_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:

lc (nm)	(lmax - lmin) (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 l_c is known, a, b_H, and L_H may be determined as above. If l_n is known, the b_ln may be obtained from the Grating Equation (1-1). Then

H_Bln = L_H tan (b_H - b_ln) (5-9)

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.

Section 4

Table of Contents

Section 6