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:

• The focal plane may be tilted by an angle, γ. Therefore, the pixel position normally occupied by the exit slit may NOT mark the normal to the focal plane.
• The dispersion and image magnification may vary over the focal plane.
• 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.

Fig. 61a illustrates a tilted focal plane that may be present in Czerny-Turner monochromators. In the case of aberrationcorrected holographic gratings, γ, β_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 γ. This is most easily achieved by taking a well-known spectrum and iteratively substituting incremental values of ± γ, until the wavelength appearing at each pixel corresponds to calculated values.

The terms used below are consistent for aberration-corrected holographic concave gratings as well as Czerny-Turner and Fastie-Ebert spectrometers.

λ_C - Wavelength (in nm) at center of array (where exit slit would usually be located)
L_A - Entrance arm length (mm)
L_Bλn - Exit arm length to each wavelength located on the focal plane (mm)
L_Bλc - Exit arm length to λc (CZ and FE monochromators L_A = L_Bλc = 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)
β_H - L_H  - Angle from LH to the normal to the grating (this will vary in a scanning instrument)
β_λn - Angle of diffraction at wavelength n
β_λc - Angle of diffraction at center wavelength
HB_λn - Distance from the intercept of the normal to the focal plane to the wavelength λ_n
HB_λc - Distance from the intercept of the normal to the focal plane to the wavelength λ_c
P_min - Pixel # at extremity corresponding to λ_min (e.g., # 1)
P_max - Pixel # at extremity corresponding to λ_max (e.g., # 1024)
P_w - Pixel width (mm)
P_c - Pixel # at λ_c (e.g., # 512)
P_λ - Pixel # at λ_n
γ - Inclination of the focal plane measured at the location normally occupied by the exit slit, λ_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 γ = 0° in one case and γ ≠ 0° in the other.

Case 1 γ = 0°

See Fig. 61b.
L_H = L_B = F at λ_c (mm)
β_H = β at λ_c
HB_λn = P_w (P_λ - P_c) (mm)

HB is negative for wavelengths shorter than λ_c.
HB is positive for wavelengths longer than λ_c.

(70) β_λn = β_H - tan^-1(HB_λn/L_H)

Note: The secret of success (and reason for failure) is frequently the level of understanding of the sign convention. Be consistent and make reasonably accurate sketches whenever possible.

To make a calculation, α and β at λ_c can be determined from Equations (2), (19). At this point, the value for α is used in the calculation of all values β_λn for each wavelength.

Then,

Case 2: γ does not equal 0°

See Fig. 61a).

L_H = cos_γ (where F = LB_λc)

β_H = β_λc + γ

HB_λc = Fsin_γ

HB_λn = P_w(P_λ - P_c) + HB_λc

λ_n = β_H - tan^-1(HB_λn / L_H)

Again, keeping significant concern for the sign of HB_λn, proceed to calculate the value B_λn after first obtaining α at λ_c then use Equation (71) to calculate λ_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 a Czerny-Turner monochromator used in spectrograph mode.

L_B = 320 mm at λ_c = F
n = 1800 g/mm
D = 24°
L_H = 319.719 mm
γ = 2.4°
HB_λc = 13.4 mm
Array length = 25.4 mm; λ_c appears 12.7 mm from end of array
λ_min, λ_max = wavelength at array extremities
λ_{error min, max} = wavelength thought to be at array extremity if γ = 0°
Disp = dispersion (Equation (5)) (nm/mm)
mag = magnification in dispersion plane (Equation (32))
Δλ(γ = 0°) = λ_min or λ_max - λ_error (nm)
Δd = Actual distance of λ_error from extreme pixel (μm)

Examination of the results given in the worked example indicates the following phenomena:

If an array with 25 μm pixels was used and the focal plane was assumed to be normal to λ_c rather than the actual 2.4°, at least a one pixel error (32 μm) would be present at λ_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).

A 25 μm entrance slit is imaged in the focal plane with a width of 27.25 μm (1.09 × 25) at 229.946 nm (when λ_c = 250 nm) but is imaged with a width of 37.75 μm at 713.2nm (1.51 × 25) (when λ_c = 700 nm), Indeed in this last case the difference in image width at λ_min compared to λ_max varies by over 10% across the array.

If the array did not limit the resolution, then a 25 μm entrance slit width would produce a bandpass of 0.04 nm. Given that, in the above example with γ = 0° rather than 2.4°, the wavelength error at λ_min exceeds 0.04 nm. Therefore, a spectral line at this extreme end of the spectral field could “disappear” the closer λ_c comes to the location of the exit slit.

The spectral coverage over the 25.4 mm array varies in the examples and are calculated as follows:

In this case, provided λ_c is known, α, β_H, and L_H may be determined as above. If λn is known, the β_λn may be obtained from the Grating Equation, Equation (1). Then

(72) HBλ_n =L_H tan (β_H - β_λn)

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.