Subsections

• 4.2.1 Grating
• 4.2.2 Fabry-Pérot

4.2 Wavelength Models

4.2.1 Grating

For each grating section, rotating scan mirrors were used to vary the angle of incidence onto the gratings. The relation between the incident angles and the resulting reflected wavelength is given by the grating equation:

$$\frac{\lambda\times N}{D} = \sin(\theta + \beta) + \sin (\theta + \delta)$$ (4.1)

where $\beta$ is the angle from the optical axis to the entrance or light source used, as seen from the collimator; $\delta$ is the angle from the optical axis to the detector pixel; $\theta$ is the angle of incidence on the grating for a beam entering along the optical axis for the current scanner position; $D$ is the grating constant; and $N$ is the spectral order.

The angle of the scan mirror, $\theta$, is measured as a `grating position' by a Linear Voltage Differential Transducer (LVDT) and colloquially given the units `LVDT'. As a first approximation the relation between scan angle and LVDT reading is well represented by a linear relation. However, the high wavelength calibration accuracy required modelling of the non-linear residuals in the $\theta$-LVDT relation. It was found from the ILT calibration that this relation could be adequately described by a fifth order polynomial function. Eighteen constants were needed to be determined for a full description of the wavelength calibration: 6 for the polynome, 3 for the aperture offsets, 6 for the detector arrays and 3 for the offset angles to the grating and FP wavelength calibration sources. The offset angles to the detector bands were described for each array with an offset to the first array elements plus a constant pitch value between the elements.

In ground-based instrument level tests, the relation between grating position readout and physical grating angle was determined by measurement of wavelength references in the form of vapour absorption lines (H$_2$O, NH$_3$, HCl). The spectral features provided by the internal grating wavelength calibrator have been tied to that scale.

4.2.2 Fabry-Pérot

A Fabry-Pérot etalon transmits only those wavelengths which fit a whole number of times within the gap between the etalon meshes.

$$\lambda = d / N$$ (4.2)

where $d$ is the (optical) gap width and N is an integer number (typically a few thousands). The optical gap is related to the mechanical gap which in turn is derived from the FP position as is explained in Section 7.2.11.2.

For the Fabry-Pérot, the position-gap relation has been determined from the spectrum of the internal FP wavelength calibrator, which is known to high accuracy from Fourier Transform Spectroscopy. H$_2$O and NH$_3$ vapour absorption lines have been used for additional checks and for determination of the variation of effective FP gap with wavelength.

In orbit, the grating position-angle relation and the FP position-gap relation was re-established during the PV phase, first using the internal calibrators and then astronomical sources. It was then checked regularly during the mission.