## Optics Tutorial

# 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:

- 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.

Figure 21(a) 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 CzernyTurner (CZ) and FastieEbert (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 ± γ, until the wavelength appearing at each pixel corresponds to calculated values.

## 5.1 The Determination of Wavelength at a Given Location on a Focal Plane

The terms used below are consistent for aberrationcorrected holographic concave gratings as well as CzernyTurner and FastieEbert 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 lc (CzernyTurner and FastieEbert 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} - Angle from L_{H} 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

H_{Bλn} - Distance from the intercept of the normal to the focal plane to the wavelength λ_{n}

H_{Bλc} - 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 CzernyTurner type instruments where γ = 0° in one case and γ ≠ 0° in the other.

**Case 1 γ = 0°.**

See Figure 21(b).

L_{H} = L_{B} = F at λ_{c} (mm)

β_{H} = β at λ_{c}H_{Bλn} = P_{w} (P_{λ} - P_{c}) (mm)

H_{B} is negative for wavelengths shorter than λ_{c}.

H_{B} is positive for wavelengths longer than λ_{c}.

(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.

To make a calculation, α and β at λ_{c} can be determined from Equations 1-2 and 2-1. At this point the value for α is used in the calculation of all values β_{λn} for each wavelength.

Then

(5-2)

**Case 2: γ does not equal 0°**

See Figure 21(a).

L_{H} = F cos γ (where F = L_{Bλc}) (5-3)

β_{H} = β_{λc} + γ (5-4)

H_{Bλc} = F sin γ (5-5)

H_{Bλn} = P_{w} (P_{λ} - P_{c}) + H_{Bλc} (5-6)

β_{λn} = β_{H} - tan^{-1} (H_{Bλn} /L_{H}) (5-7)

Again keeping significant concern for the sign of H_{Bλn}, proceed to calculate the value β_{λn} after first obtaining α at λ_{c} then use Equation (5-2) 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 CzernyTurner 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°

H_{Bλc} = 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 1-5) (nm/mm)

mag = magnification in dispersion plane (Equation 2-16)

Δλ(γ = 0°) λ_{min} or λ_{max} - λ_{error} (nm)

Δd = Actual distance of λ_{error} from extreme pixel (μm)

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

nm | λ_{min}229.9463 |
λ_{c}250 |
λ_{max}269.7469 |
λ_{min}381.4545 |
λ_{c}400 |
λ_{max}418.1236 |
λ_{min}686.1566 |
λ_{c}700 |
λ_{max}713.1999 |
---|---|---|---|---|---|---|---|---|---|

α | 1.29864 | 1.29864 | 1.29864 | 9.5950 | 9.5950 | 9.5950 | 28.0963 | 28.0963 | 28.0963 |

β_{H} |
27.6986 | 27.6986 | 27.6986 | 35.9950 | 35.9950 | 35.9950 | 54.496 | 54.496 | 54.496 |

β | 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 |

Δλ | 0.051 | 0 | 0.015 | 0.048 | 0 | 0.014 | 0.037 | 0 | 0.011 |

Δd | +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 lc 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).

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 λ_{c} = 250 nm) but is imaged with a width of 37.75 µm at 713.2 nm (1.51 x 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.

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 γ = 0° rather than 2.4°, the wavelength error at lmin 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.

D. The spectral coverage over the 25.4 mm array varies in the examples calculated as follows:

λ_{c}(nm) |
(λ_{max} - λ_{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 λ_{c} is known, α, β_{H}, and L_{H} may be determined as above. If λ_{n} is known, the β_{λn} may be obtained from the Grating Equation 1-1. Then

H_{Bλn} = L_{H} tan (β_{H} - β_{λn}) (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.