where 6(k — k0) is the Dirac delta function, and thus the reconstructed spectrum is found to be purely monochromatic as expected. However, in reality the pathlength 2^ may only be scanned up to a maximum value of Am, and this limits the spectral resolution of the interferometer. If we limit A to Am in Equation (7.8), then the reconstructed spectrum is no longer monochromatic, but instead has finite width since
Since k0 is typically large (except for the very long wavelengths) this may usually be approximated as
and thus the reconstructed spectrum has a finite spread of wavelengths, although this becomes more monochromatic as Am is increased since I(k) ^ S(k — k0) as Am ^ as expected. Hence, an interferogram which is truncated with a maximum path difference of Am has an effective spectral resolution of = 1/Am. Because of this feature, the spectral resolution of a Fourier Transform Spectrometer is easily adjustable by simply recording longer or shorter interferograms, and thus the same instrument may record spectra with multiple resolutions. In addition, the shape of the spectral instrument function may also be adjusted through the process of a^odi'zan'ora. We can see in Equation (7.10) that the instrument function, when an interferogram is transformed directly with a hard cut-off at Am, is essentially a sinc function. Suppose that there was a weak feature in the true spectrum very close to a strong feature. Because the sinc function has non-negligible ripples next to it, the weak feature might easily get lost in the ripples of the strong feature in the reconstructed spectrum, especially if noise was also present. To avoid this, the interferogram may first be multiplied by an apodizing function ^(A), which instead of imposing a hard cut-off at Am forces the interferogram to decay smoothly to zero at Am. It may easily be shown that this changes the effective shape of the instrument function and if applied correctly may completely remove the ripples from strong features at the expense of slightly lowering the overall spectral resolution. This technique is thus called apodization which derives from Greek words meaning literally "removal of feet"!
Another factor to consider in real Fourier Transform Spectrometers is aliasing. In practice an interferogram is sampled at a finite resolution of the path difference A = 2x. Higher frequencies in the observed spectrum can be seen from Equation (7.5) to appear as higher and higher frequency components in the interferogram. If the minimum sampling path difference is As, then the maximum frequency in the incident spectrum that may unambiguously be reconstructed, according to the Nyquist sampling theorem, is v0 = 1/(2AS) (James and Stern, 1969; Vanasse, 1983). Hence, if any higher frequencies are detectable by the system then they may artificially appear at lower and incorrect frequencies between 0 and This phenomenon is known as aliasing and may be removed by ensuring that the frequency response of the actual detection system is limited to the frequency range defined by the sampling limit.
Together with allowing variable instrument functions, one of the major advantages of interferometers is that they simultaneously record data over a wide wavelength range and thus little radiation that is collected by the telescope system is wasted. This property also means that interferometers suffer less from FOV variations that may occur during an interferogram scan than grating spectrometers. One possible drawback of the classic Michelson interferometer design for space operation is that the flat mirrors need to be very precisely aligned in order that the central spot of the interference pattern falls on the detector. Such a precise alignment can easily be destroyed during the vibrations which accompany the launch of spacecraft, although the IRIS instruments on Voyager (and previous Earth and Mars missions), which had simple Michelson designs, were successful. More recently, the CIRS interferometer on the Cassini spacecraft (Section 7.9.2) uses corner reflectors, and roof reflectors, which are much less sensitive to misalignment.
The microwave emission of the giant planets is extremely weak and difficult to detect, but technology is rapidly improving. Two main types of receivers are currently used: (1) bolometers; and (2) heterodyne receivers.
With bolometers the radiation is again collected in a tiny absorber, whose temperature changes are converted to electrical signals, which are then amplified and measured. Bolometers can detect broadband radiation (e.g., within the microwave atmospheric windows described in Section 7.3.1), with high sensitivity, but cannot give information on detailed spectral energy distribution within that band.
Heterodyne receivers of microwave radiation operate by first converting the microwave signals to a lower frequency by nonlinear mixing with a local oscillator signal. The converted lower frequency signal may then be amplified and measured with conventional radio frequency (RF) electronics. Such devices have very high spectral resolution (of the order of 106) and so can examine individual absorption lines.
The two critical components of such receivers are the local oscillator and the heterodyne mixer. The local oscillator defines the spectral frequency observed and should be very stable, have low noise, and ideally have low power. The heterodyne mixer allows the measured spectrum to be converted to lower frequencies and be analyzed with conventional RF electronics. The most sensitive receivers at present use the strong heterodyne mixing provided by superconductor-insulator-superconductor (or SIS) tunnel junctions. An SIS junction consists of two superconducting electrodes separated by a very thin insulating barrier. Electrons tunneling across this barrier give rise to a very nonlinear current-voltage characteristic, which is the key to heterodyne mixing. Such junctions need to operate at very cold temperatures in order to achieve superconductivity depending on the material used in their construction, typically the temperature of liquid helium (4.2 K). As an example the SIS junctions currently used by IRAM (Section 7.6.1) consist of a superposition of a thin layer (of the order of 4 ^m) of aluminum oxide between two layers of the superconducting metal niobium. However, SIS junctions are still very much under development and thus a number of different superconducting alloys and insulating layer materials are currently under investigation all over the world. In addition to SIS junctions, other types of mixers are being studied, such as Schottky diodes, which can operate at higher temperatures and this is an area of rapid technological development.
Apart from low-noise characteristics, the main advantage of heterodyne receivers is their capability to provide high-resolution spectroscopy, which is very important for detecting the absorption lines of microwave absorbers such as CO and HCN, and determining their abundance by accurately measuring their line depth.
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