Grating Operation

Radiation entering the LWS was dispersed by the grating so that different wavelengths were transmitted at different angles. The grating was set at an angle of $60\hbox{$^\circ$}$ to the incident beam but could be scanned by up to $\pm 7\hbox {$^\circ $}$. It was a blazed reflection grating with blaze angle of $30\hbox{$^\circ$}$ and 7.9 lines per mm (Clegg et al., 1996).

The detector block was set at a fixed angle to the incident beam so that as the grating was scanned the wavelength transmitted to each detector changed (see Figure 1.4). The use of a blazed grating meant that the peak intensity in the grating diffraction pattern was shifted out of the undispersed zeroth order into a more useful higher order (e.g. Hecht, 1987). With a blaze angle of $30\hbox{$^\circ$}$ and incident angle of $60\hbox{$^\circ$}$, the peak was shifted to be at an angle of $0\hbox{$^\circ$}$ to the normal rather that in the zeroth order at $60\hbox{$^\circ$}$. This meant that maximum energy was transmitted to the detectors which were operated in grating first and second order.

The wavelength range (in $\mu$m) that was transmitted to each detector by scanning the grating angle can be calculated from the following equation,

$$\frac{sin(\theta_{i})-sin(\theta_{det}-\theta_{i})}{N}=m\lambda$$ (A.1)

where $m$ is an integer representing the order of interference, $N$ is the number of lines per micron (7.9$\times10^{-3}$ $\mu$m$^{-1}$), $\theta_{i}$ is the angle between incident beam and grating normal ( $60\pm7\hbox{$^\circ$}$) and $\theta _{det}$ is the angle between the incident beam and each detector. $\theta _{det}$ was set by the geometry of the instrument and the value for each detector is shown in Table 1.2. This table also shows the wavelengths reaching each detector when the grating angle was scanned by up to $\pm 7\hbox {$^\circ $}$. To limit the detected radiation to a single grating order, band-pass filters were placed over each detector, defining ten overlapping wavelength bands. These filters set the nominal operating range for each detector. Short wavelengths were covered by detectors labelled SW1-SW5 in approximately 15 $\mu$m wide channels. Longer wavelengths were covered by detectors LW1-LW5 in approximately 30 $\mu$m wide channels (see Table 1.2). Detectors SW1 to SW5 were always operated in grating second order and detectors LW1 to LW5 were always operated in grating first order.

Table 1.2: Detector types (`s' represents the stressed detectors), angles between the beam incident on the grating and each detector ($\theta _{det}$) and the wavelength ranges transmitted to each detector by scanning the grating by $\pm 7\hbox {$^\circ $}$. The final nominal operating range for each detector, defined by its band-pass filter, is also shown.

Detector	Type	$\theta _{det}$	Grating	Wavelength Range at $\theta _{det}$	Nominal Range
Name			Order	($\mu$m)	($\mu$m)
SW1	Ge:Be	$67.800\hbox{$^\circ$}$	2	34.38-57.38	43.0-50.5
SW2	Ge:Ga	$58.740\hbox{$^\circ$}$	2	44.22-67.35	49.5-64.0
SW3	Ge:Ga	$49.710\hbox{$^\circ$}$	2	54.18-77.07	57.0-70.0
SW4	Ge:Ga	$40.730\hbox{$^\circ$}$	2	66.00-86.27	67.0-82.0
SW5	Ge:Ga	$31.720\hbox{$^\circ$}$	2	73.52-94.82	76.0-93.0
LW1	Ge:Ga	$63.260\hbox{$^\circ$}$	1	78.55-124.78	84.0-110.0
LW2	Ge:Ga (s)	$54.290\hbox{$^\circ$}$	1	98.24-144.37	103.0-128.0
LW3	Ge:Ga (s)	$45.270\hbox{$^\circ$}$	1	118.12-163.39	123.0-152.0
LW4	Ge:Ga (s)	$36.275\hbox{$^\circ$}$	1	137.52-181.19	142.0-171.0
LW5	Ge:Ga (s)	$27.320\hbox{$^\circ$}$	1	155.95-197.34	161.0-196.0

The spectral resolving power of an ideal grating is determined by the number of illuminated rulings and the order of interference,

$$\frac{\lambda}{\Delta\lambda_{\mathrm{g}}}=mNl$$ (A.2)

where $\Delta\lambda_{\mathrm{g}}$ is the least resolvable wavelength difference (defined by the Rayleigh criterion to be the distance between maximum and first minimum in the transmission profile) and $l$ is the width of the illuminated region on the grating. The grating used in the LWS was oversized so that the number of illuminated lines was determined solely by the diameter of the beam footprint. This footprint was elliptical due to the angle of the incident radiation and its width depended on the scan angle of the grating, $\theta_{\mathrm{g}}$, and on the spatial extent of the beam (which was wavelength dependent due to diffraction in the optical system; Lloyd, 2002c). Equation 1.2 shows that the resolving power of a grating increases for higher values of $m$. The angular dispersion of a grating also increases with the order. This is defined to be the change in angular position corresponding to a certain difference in wavelength and is given by (e.g. Hecht, 1987),

$${\cal{D}}=\frac{mN}{\cos{(\phi)}}$$ (A.3)

where $\phi$ is the angle to the grating normal. In the LWS this was equal to $(\theta_{det}-\theta_{i})$ - see Figure 1.4. Equation 1.3 shows that for each order the dispersion of a grating increases with the angle from the grating normal.

The average FWHM of the grating response function was measured on the ground and found to be 0.6 $\mu$m for first order and 0.29 $\mu$m for second order. This gave a resolving power of between 150 and 300. The shape of the response function was determined from calibration observations of unresolved lines made in orbit. The profile was found to be stable with time and line strength and there were no significant differences between detectors. There was also no distortion in shape found for lines observed towards extended sources (Gry & Lorente, 2000). The line widths measured in orbit were found to be slightly wider than those measured on the ground but this could have been due to detector transient effects^A.2. The grating response for first and second order are shown in Figure 1.5. The shapes are not Gaussian but can be approximately fitted with a Gaussian profile. This introduces an error in the integrated line flux of the order of 2% (Gry & Lorente, 2000).

Figure 1.51.5: Grating response functions for first order (detectors LW1-LW5) and second order (detectors SW1-SW5) from C. Gry (private communication).