**Figure 2.4:** The side view shows an incident beam (red line), diffracted by the grating at its rest position (solid blue line, limits of rotation shown by dashed blue line). The shorter wavelength radiation went to SW1 and the longer to LW5, via the spherical condensing mirror. The middle panel indicates the elliptical footprint of the beam. The bottom diagram shows the angle of the incident beam to the normal. The diffracted beam was also normal to the grating (for one wavelength), in its rest position, because a blaze angle of 30 $^{\circ}$ was chosen for optimum performance.
$\resizebox {11.5cm}{!}{\includegraphics{grating2.eps}}$

In its rest position, the grating normal was at 60 $^{\circ}$ to the incident beam (see Figure 2.4). By rotating the grating, by means of a servo-controlled drive mechanism, between $-7^{\circ}$ and $+7^{\circ}$ (the physical limits of its motion), the centre of the grating response function was scanned over a wavelength band, for SW4 this ranged from 64 to 86 $\mu$ m; it is shown in the upper panel of Figure 2.5. Considering each detector in turn, a contiguous coverage from 45 to 180 $\mu$ m was achieved by rotating the grating from $-3.5^{\circ}$ to $+3.5^{\circ}$ . By using the extended range of operation ( $-7^{\circ}$ to $+7^{\circ}$ ) -- at the cost of a small increase in power dissipated in the focal plane -- the spectral range was extended to cover 43-197 $\mu$ m, whilst giving significant overlap between the spectral coverage of adjacent-wavelength detector channels. In order to maximise the sensitivity of the instrument at all wavelengths, the extended scanning range has been used in normal operation. The wavelength ranges used for each detector, along with the overlaps, are indicated in the upper panel of Figure 2.5 (note that the limitations on the detector wavelength ranges, as discussed below, are taken into account). The nominal well calibrated ranges are shown in yellow. The extended range gave important verification when looking for weak line features and afforded redundancy in the instrument if there had been a catastrophic failure in one detector.

**Figure 2.5:** The top plot shows the range covered by each of the ten detectors, by rotating the grating to its extremes, along with the grating order and the FP wheel used. The yellow regions indicate the nominal wavelength ranges. The second plot shows the second order grating spectral element (red) at a grating angle of $-1.36^{\circ}$ . It also shows the adjacent orders of this radiation and the detector filter which prevented these orders from reaching the detector. The final plot shows the n $^{th}$ order Airy profile of the FP with the superimposed grating spectral element, also the range of wavelengths which could be covered by changing the mesh separation is indicated by the green arrows. The neighbouring peaks that fell outside of the grating response are indicated.
$\resizebox {11.5cm}{!}{\includegraphics{big_plot.eps}}$

The grating diffracts radiation of wavelength $\lambda$ in first order at the same angle as wavelength $\lambda /2$ in second order and as wavelength $\lambda /n$ in $n^{th}$ order. With this constraint it was not possible to utilise the full wavelength coverage afforded by the range of possible scan angles. To ensure that only the required narrow band of wavelengths is detected at a particular grating angle, and not the wavelengths in different orders, filters with well-defined passbands were placed in front of the detectors. The transmission of the filters, measured by Ade (private communication), is shown in Figure 2.6. The resulting nominal wavelength limits for each band are given in Table 2.3. They take into account the following limitations:

**Figure 2.6:** *The LWS flight model filters measured by Ade (private communication). They are in sequential order SW1 to LW5 (from right to left), see also Table 2.1.*
$\resizebox {13cm}{!}{\includegraphics{this_iso_fm_filters.eps}}$

**Figure 2.7:** *The LWS flight model detector responses measured by Ade (private communication). LW5 is shown by the blue curve, LW4 red, LW3 green, LW2 cyan, SW2-LW1 yellow and SW1 magenta.*
$\resizebox {13cm}{!}{\includegraphics{iso_fm_detectors.eps}}$

The central panel in Figure 2.5 shows the nominal wavelength range covered by SW4, as determined by its filter. When the grating was in its rest position, the central wavelength falling on this detector was 75 $\mu$ m. With the grating at an angle of $-1.36^{\circ}$ its spectral response function (red) fell at 73.5 $\mu$ m. The lower panel shows the range of the wavelengths that fell on the detector at this angle, as given by the grating spectral element (red).

**Table 2.1:** Theoretical and measured LWS detector aperture sizes in the dispersion direction, Y, and the non-dispersion direction, X. Also the serial numbers of the detectors, bandpass filters and edge filters used on the LWS are listed.
Det	Y		X		Detector	Bandpass	Edge Filter
	Measured	Design	Measured	Design	Serial	Filter Serial	Serial
	[ $\mu$ m]	[ $\mu$ m]	[mm]	[mm]	Number	Number	Number

SW1	494	500	1.34	1.29	7	325	8
SW2	529	530	1.32	1.29	9	321	50
SW3	529	540	1.30	1.29	13	319	6
SW4	617	610	1.34	1.29	11	313	12
SW5	640	650	1.35	1.29	12	309	9
LW1	580	570	1.34	1.29	10	306	2
LW2	617	620	1.35	1.29	55	302	1
LW3	652	650	1.34	1.29	56	328	5
LW4	700	690	1.32	1.29	58	296	4
LW5	758	750	1.30	1.29	57	292	3

The re-imaged size of the beam,

, at the detector array was determined by the focal ratios of the collimator ( $f_{col}=15$ ) and the condenser ( $f_{con}=1.5$ in the dispersion direction and $f_{con}=3$ in the non-dispersion direction as dictated by the anamorphic magnification, see below) and the size of the focal plane aperture, M2 (

) such that:

Equation 2.1 gives the aperture size in the dispersion direction to be 0.465mm. However, there is a modification of the beam cross-section which is referred to as anamorphic magnification (AMAG; the ratio of the diameter in the non-dispersion direction to that in the dispersion direction). It occurred because the radiation was not specularly reflected with respect to the plane of the grating, hence the emergent beam was elliptical. (The incident beam was circular and it made an elliptical footprint on the grating, but this was due to purely geometrical effects.) This AMAG reduced the image size such that a detector aperture of 0.7mm was actually required for the dispersion direction.

AMAG was smallest for the most positive scan angles (long wavelength end) of each detector range, so to ensure good efficiency the positive scan angle limit was used to determine the beam size and consequently the aperture size for each detector in the dispersion direction. The final measured and designed aperture sizes for the detectors are given in Table 2.1 along with the serial numbers of the actual detectors, bandpass filters and edge filters that flew on ISO.

For the LWS the AMAG is typically equal to two. As the parallel beam had a width of 34mm, the beam was typically dispersed over 68mm. The grating was ruled with 7.9 lines per mm, hence the number of lines covered by the beam in the dispersion direction was $\sim$ 540. For the above configuration, the chromatic resolving power (

, where

is the order and

is the number of lines) is $\sim$ 1080 in second order and $\sim$ 540 in first. However, in reality the chromatic resolution depends on the detector location (as different detectors view the grating at different angles) and also on the scan angle of the grating. The LWS beam size is wavelength dependent. An effective aperture radius for each detector has been defined by Lloyd 2000, [27] (see Section 5.9) and is listed in Table 5.9.

This array was therefore capable of simultaneously detecting ten spectral elements within the LWS spectral region. However, the packing density of the detectors was sparse (limited by the size of the detector mounts), so to get complete spectral coverage the grating had to be scanned to move the wavelengths sequentially across each detector. By having ten detectors rather than one, the whole spectrum could be obtained in a tenth of the time. Because of the wide spectral coverage of the LWS, it was necessary to use the grating in first order for wavelengths from 94.6-196.9 $\mu$ m and second order for wavelengths 43-94.6 $\mu$ m, to maximise its efficiency. The grating efficiency measurements, as performed by Petti 1989, [31], are shown in Figure 2.8. Because of the two orders used in the LWS, it was necessary to interleave the long wavelength detectors between the short wavelength detectors to make optimum use of the limited space available whilst maintaining the maximum spectral range. Accordingly, the detectors are labelled SW1 though SW5 for the short wavelength set and LW1 to LW5 for the long wavelength set. The detectors in their different positions saw the grating at different angles. The diffracted beam for detector SW1 emerged at an angle of $+7.9^{\circ}$ with respect to the grating normal, as shown in the top part of Figure 2.4. A simplistic way to determine where the detectors were located in the LWS is to refer to the angle between the incident beam and the direction of the detector from the grating, as tabulated in Table 2.2.

**Figure 2.8:** *The efficiency of the LWS grating.*
$\resizebox {13cm}{!}{\includegraphics{grat_eff.ps}}$

**Table 2.2:** The angles (ground and in orbit) between the incident beam on the grating and the detectors (these angles are shown in Figure 2.4 for SW1 and LW5) and the corresponding wavelengths for the grating in the nominal position.
Detector	Detector Angles [ $^{\circ}$ ]		Wavelength [ $\mu$ m]
	Ground	In Orbit	Ground	In Orbit

SW1	67.938	67.80	46.0711	46.2220
LW1	63.411	63.26	102.092	102.425
SW2	58.889	58.74	56.0389	56.2033
LW2	54.370	54.29	122.042	122.218
SW3	49.885	49.71	65.9272	66.1173
LW3	45.340	45.27	141.659	141.809
SW4	40.825	40.73	75.6000	75.6989
LW4	36.308	36.275	160.487	160.554
SW5	31.785	31.72	84.7346	84.7977
LW5	27.256	27.32	178.090	177.971

2.4 The Grating Spectrometer