Fabry-Pérot Operation

To operate the instrument at high resolution, two FP interferometers could be added into the beam in front of the grating. An exploded view of the FP mounting is shown in Figure 1.6. The design of the LWS FP system was described in detail by Davis et al. (1995). An ideal FP interferometer consists of two highly reflecting parallel plates. Rays entering the FP are reflected many times within the gap between the plates. Rays finally emerging from the FP have a change in phase due to the path length difference from reflection in the gap. A FP spectrometer works because the phase change, $\delta$, is a function of wavelength (e.g. see Hecht, 1987),

$$\delta = \frac{4\pi n_{f}}{\lambda} d \cos{(\theta_{i})} + 2\psi$$ (A.4)

where $n_{f}$ is the refractive index of the medium, $\lambda$ is the wavelength of incident radiation, $d$ is the gap between plates and $\theta_{i}$ is the angle of incident radiation. $\psi$ is an extra phase change due to absorption in the plate material.

Figure 1.61.6: Exploded diagram of one of the FP interferometers, showing the mounting and mechanism for scanning the gap between the meshes.

When multi-wavelength radiation is incident on a FP interferometer, the relative transmitted flux density distribution is an infinite series of discrete peaks in transmission, given by the Airy function,

$$\frac{I_{t}}{I_{i}}=\frac{1}{1+F\sin^{2}{(\delta/2)}}$$ (A.5)

where $F$ is the coefficient of finesse of the FP, defined by the reflectance of the FP plates, $R$, to be,

$$F=\left(\frac{2\sqrt{R}}{1-R}\right)^{2}$$ (A.6)

The reflectance also determines the FWHM of the peaks, $\Delta\lambda_{\mathrm{f}}$,

$$\Delta\lambda_{\mathrm{f}} = \frac{1-R}{\pi\sqrt{R}} = \frac{2}{\pi\sqrt{F}}$$ (A.7)

The spectral resolving power is defined to be,

$$\frac{\lambda}{\Delta\lambda_{\mathrm{f}}} = \frac{\pi\lambda\sqrt{R}}{1-R} = \frac{\pi\lambda\sqrt{F}}{2}$$ (A.8)

Peaks in transmission occur when $\delta$/2 = $m\pi$, where $m$ is an integer representing the order of the peak. In the LWS, the FP gap could be scanned to change the wavelengths transmitted and changing $\delta$ by 2$\pi$ (i.e. moving from one order to the next) corresponded to a difference in wavelength of,

$$\Delta\lambda_{\mathrm{fsr}}=\frac{\lambda}{m}$$ (A.9)

known as the free spectral range.

The reflectance and absorptance of the plates defines the overall transmission efficiency for a FP and this determines the measurement sensitivity that can be achieved. The transmission efficiency is given by,

$$\eta=\left(1-\frac{A}{1-R}\right)^{2}$$ (A.10)

where $A$ is the absorptance and $R$ is the reflectance of the plates. In order to achieve both high spectral resolution and high efficiency, the absorptance and reflectance must satisfy $A \ll 1-R \ll 1$ across the desired spectral range. Due to the difficulty of simultaneously meeting these two requirements in practice, there is a trade off between resolution and efficiency. In the LWS this problem was resolved by using two FP interferometers to cover the full range. High reflectance and low absorptance were achieved by using metal meshes as the FP plate material (Davis et al., 1995).

Equation 1.4 can be simplified for the LWS because the surrounding medium was a vacuum (with refractive index of 1.0) and the FP selection wheel was placed so that the beam was always at normal incidence to the FP meshes. This meant that peaks in transmission occurred when,

$$\frac{4\pi}{2\lambda}d=m\pi$$ (A.11)

or,

$$2d=m\lambda$$ (A.12)

This assumes that there was no extra phase shift due to absorption in the FP plate material. No evidence for such an extra phase shift has been found in the LWS when the FPs are operated within their nominal wavelength ranges.

Figure 1.7: FP transmission for orders at wavelengths from $\lambda _{2}$ to $\lambda _{-3}$. The order at $\lambda _{0}$ moves across the grating spectral response profile (dotted line) creating one mini-scan.

In order to actually use a FP spectrometer, it is necessary to select a single FP order from the infinite series of transmission peaks. In the LWS this was achieved by using the grating response profile as an order selector. If the FP orders were sufficiently far apart, only a single FP order was transmitted at each grating angle (see Figure 1.7). Normally two FP interferometers are used in tandem to carry out order selection and it is rare to use a grating for this purpose. This instrumental set-up was designed to make it easy to change between the FP and grating modes whilst minimising the space required for the FP system. There were also some very useful side-effects that only became apparent when the instrument was actually used. These are described in more detail in Section 1.6 and meant that useful data were sometimes recorded on non-prime detectors.

In order to record data over a range in wavelength, the FP gap was scanned. This shifted the wavelength of the selected FP order across the grating response function. Data were recorded at each FP scan position, creating one `mini-scan' (see Figure 1.7). A large range in wavelength could be covered by changing the grating angle and repeating the FP gap scan. This created a sequence of mini-scans. The resolving power of the instrument was increased by inserting the FP into the beam to $\sim$6600-7900 for FPS and $\sim$6000-10000 for FPL, giving a resolution element of 30-40 km s$^{-1}$. The actual resolving power measured on the ground was lower than expected from the design model predictions and varied with wavelength (Davis et al., 1995). This was initially attributed to the transient response of the detectors but subsequent measurements in flight confirmed the wavelength dependent behaviour. The best determination of the resolving power from ground and space based measurements is shown in Figure 1.8 (a) (B. M. Swinyard, private communication).

Figure 1.81.8: (a) Resolving power of FPS (blue) and FPL (red) from ground-based and in-flight measurements. The solid lines are fits to the data (spline fit to FPS and polynomial fit to FPL; B. M. Swinyard, private communication). (b) Total measured FP throughput for FPS and FPL, defined as the combination of FP efficiency and resolution element width.

The transmission efficiency of each FP was difficult to measure independently of the resolution element. To solve this problem in the calibration procedure, the total measured FP `throughput' was defined to be a combination of efficiency and FP resolution element width as this was much easier to measure in practice. The variation in FP throughput with wavelength is shown in Figure 1.8 (b) for FPS and FPL and its derivation is described in more detail in Section 3.6.4.

One problem with FP spectrometers is that the free spectral range (equation 1.9) decreases with decreasing wavelength. For most wavelengths in the LWS range, the grating response was sufficiently narrow to select a single FP order. However, towards the short wavelength end of the range over which each grating order was used, more than one FP order were sometimes included (see Figure 1.7). This caused the measured flux to be too high. This effect is known as FP side order contamination and is described in more detail in Section 3.6.

The shape of the FP spectral response function was determined from dedicated calibration observations of narrow, unresolved spectral lines in planetary nebulae spectra. The FP gap was slowly scanned backwards and forwards across the line to obtain high signal to noise. This slow scanning mode allowed the detectors to settle between measurements and avoided any distortion in the shape from transient memory effects in the detectors. These observations showed that the response function shape could be accurately described by the theoretical Airy profile (equation 1.5) with the coefficient of finesse set by the measured resolving power (B. M. Swinyard, private communication). Figure 1.9 shows the ionised carbon line at 157.7 $\mu$m and the atomic oxygen line at 63 $\mu$m measured in the spectrum of NGC 7023. The theoretical Airy profile is over plotted and this shows a good fit to the data. No asymmetries were observed in the line profile shape.

Figure 1.9: A measurement of the [CII] 157.7 $\mu$m line using FPL order $m=64$ and the [OI] 63 $\mu$m line using FPS order $m=86$ in planetary nebula NGC 7023 (B. M. Swinyard, private communication). This was measured in a dedicated calibration observation using the FP in a slow scanning mode. The theoretical Airy profiles at these wavelengths are overplotted and the residuals are shown below.