where B is the radiance from the source, As is the area of either slit, and e(A) is the throughput. The solid angle ^ is equal to the area of the grating, Ag, divided by the square of the focal length f of the collimator and condenser. The flux can be rewritten in terms of the dispersion at the exit slit, da/dA, thus
So, the flux obtainable at a given resolution is improved by minimizing the ratio of focal length f to slit height h (f-number) and the area and dispersive power of the grating. Optical-design considerations generally limit the /-number to about unity, and the dispersion to < 1 mr nm_1. Then, the only means remaining to improve the signal-to-noise ratio for a given source is to use larger cross-section optics.
The general problem of obtaining adequate signal-to-noise ratio with instruments viewing the Earth from space can be illustrated with some rough order of magnitude numbers. Suppose that the grating in this example has an area of 10x10 cm2 and is 10 cm from an entrance slit of area 0.1 cm2, illuminated by a 1000 K source. These are typical order-of-magnitude numbers for real systems. Suppose further that the device has a spectral resolution of 0.01 cm-1 and that we wish to obtain the spectrum from 1000 to 10000 cm-1 with a maximum SNR of 100 using an uncooled thermistor detector with NEP = 10-10 W Hz-1/2. A straightforward calculation shows that such a scan would take on the order of one day to complete, illustrating the problem of making measurements with simultaneous high spatial and spectral resolution from a satellite that orbits the Earth more than 10 times per day. Cooling the system, and thereby gaining an improvement of 2 or more orders of magnitude in signal-to-noise ratio, becomes essential. Limiting the spectral range to those wavelengths that contain most of the desired information, for example the 667-cm-1 carbon dioxide band for atmospheric temperature sounding, further improves the practicality of the measurements. Finally, large-aperture optics are often used. These essential improvements raise the mass of space instruments making climate observations, like those discussed in the next chapter, to more than ten times that of the device described above that is typical of those used for field geology on the Earth's surface.
The principle of the diffraction grating involves optical interference, of course, but the term interferometer is generally reserved for slitless devices like the moving-mirror Michelson, where the full circular aperture can be used and a gain in throughput of up to two orders of magnitude results. The basic principle of Fabry-Perot and Michelson-type devices is to divide the incoming beam into two using a partially reflecting beamsplitter and then modulate the path difference traversed by the two components before they recombine. This is achieved in the Michelson device by moving one or both of the mirrors that return the two components of the divided beam to the beamsplitter for recombination. The use of corner cubes instead of plane mirrors to return the two beams means that any tilting of the reflector affects only the translation of the beam and has no first-order spectral effects, making the device less sensitive to alignment errors. Various schemes are employed to fold and multipass the beam in order to maximize the optical-path difference, while keeping the overall dimensions of the instrument reasonably compact, which is particularly important for use on satellites and if cooling is employed to boost the sensitivity. The maximum path difference achievable in practice is about 1 m, enabling spectra to be obtained with a resolution of 0.005 cm-1. In a practical system, provision may be made for the selection of different filters and detectors for different wavelength regions, each of limited extent to exclude source photons from parts of the spectrum not being scanned. This improves the signal-to-noise ratio by reducing the photon noise in the cooled photon detectors.
Interferometers have a number of general advantages over other types of spec-trometric instrument, including higher spectral resolution, higher throughput, and wider wavelength coverage. They are, however, prone to additional sources of noise that can annul the advantages in a realistic, as opposed to an ideal, system. The first of these is phase errors. An interferogram may not be the simple sum of sine waves that it is in theory, if the path difference inside the instrument is a function of frequency due to optical dispersion in the beam splitter or other components. Phase shifts may also be introduced in the electronics. A second problem area in interferometers is channelling, a periodic modulation of the spectral baseline caused by optical interference inside individual components, and by stray light inside the system.
The effects of detector non-linearity on an interferogram are difficult to untangle, and this type of instrument is more likely to suffer from the problem in the first place because of the large dynamic range required of the detector. Fourier spectrometers are also extremely susceptible to vibration, electromagnetic interference, and to sampling errors. In summary, the design of an instrument of this type is an extremely complex and specialized affair, which explains the enduring popularity of grating and filter instruments in spite of their inferior theoretical performance.
An introduction to the topics in this chapter may be found in the books by Taylor (Chapter 9), Harries, Stevens and Houghton et al. Beer provides an excellent overview of the interferometric technique and Chantry of infra-red techniques in general; Boyd focuses on detector theory. There is, of course, a large technical literature on the subject of satellite remote sensing instrumentation; see for instance the website of the Society of Photo-optical Instrument Engineers (SPIE).
9.8.2 References and further reading
Atkins, P. W. and Friedman, R. S. (1996). Molecular quantum mechanics. Oxford University Press, Oxford.
Beer, R. (1992). Remote sensing by Fourier transform spectrometry. John Wiley & Sons Inc., New York.
Boyd, R. W. (1983). Radiometry and the detection of optical radiation. Wiley Interscience, New York.
Chantry, G. W. (1984). Long wave optics. Academic Press, New York.
Foote, M.C. and Jones, E. W. (1998). High performance micromachined thermopile linear arrays. Proc. SPIE, 3379, 192-197. Infrared Detectors and Focal Plane Arrays V. E. L. Dereniak, R. E. Sampson (ed.) Jet Propulsion Laboratory, Pasedena.
Harries, J. E. (1994). Earthwatch: Climate from space. Wiley-Praxis Series in Remote Sensing, New York.
Houghton, J. T., Taylor, F. W. and Rodgers, C. D. (1984). Remote sounding of atmospheres. Cambridge University Press, Cambridge.
Johnson, J. B. (1928). Thermal agitation of electricity in conductors. Phys. Rev., 32, 97-109.
Nyquist, H. (1928). Thermal agitation of charges in conductors. Phys. Rev., 32, 110-113.
Schottky, W. (1918). On spontaneous current fluctuations in different electrical conductors. Ann. Phys. (Leipzig), 57, 541-567.
Steinfeld, J. I. (1985). An introduction to modern molecular spectroscopy. MIT Press, Massachusetts.
Stephens, G. L. (1994). Remote sensing of the lower atmosphere. Oxford University Press, Oxford.
Taylor, F. W. (2005). Elementary climate physics. Oxford University Press, Oxford.
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