Examples of imaging spectrometers are
Typical rays emerging from two representative points are drawn. One point is located on the optical axis. The second point is displaced by a distance y from the optical axis. The object plane is located at a distance equal to the focal length f of the collimating lens. The object plane may be the focal plane of a telescope. The camera lens, with focal length f' produces an image with magnification (or reduction) factor f/f'.
The depth of field of the focusing system should accommodate twice the maximum travel distance of the moving mirror in the interferometer in order to produce a sharp focus in the image plane. The beam splitter transmits a fraction T of the incident light and reflects a fraction R. These coefficients in general depend on polarization, the angle , and the wavelength, . For monochromatic light of wavenumber , the intensity of light at the image point y' relative to the intensity emerging from the object point y is proportional to . The phase difference is determined by the round trip optical path difference x for the moving mirror with respect to its zero phase difference (ZPD) point, and the angle , between the collimated rays and the optical axis,
For a non-monochromatic point source, the observed light intensity I(x,y') in the focal plane is a function of both mirror position x and the distance from the optical axis y'. For a spectral distribution with intensity for wave numbers between k and k+dk given by S(k)dk, and a detection efficiency E, the observed light intensity in the focal plane is given by
The light intensity at any given radius y' is thus simply related to the Fourier transform of the spectrum. Introducing the modified frequency
and the modified spectrum , the observed intensity is
Here is a simulation of the fringe pattern predicted by Eq (4).
By Fourier transforming I(x,y') as a function of x, the spectrum S'(k') can be recovered, and thus S(k) itself.
Ordinarily, a range of values of y' are integrated, which has the effect of broadening and shifting a monochromatic line in the spectrum which is recovered from the Fourier transform of I(x). This is the origin of the Jacquinot limit on the resolution of an FTS. For a Jacquinot stop of radius r, to first order in r/f, the total width of the Jacquinot blurring is
For points off the optical axis, this blurring varies linearly with distance from the optical axis. In terms of the angular position and spread of a given pixel, this limit to the spectral resolution is given by
Except for extremely high resolution instruments, this limitation is small compared to the limit on the resolution from the total travel distance of the moving mirror.
C. L., Carter, M. R., Fields, D. J. & Hernandez, J. 1993,
Proc. SPIE, 1937, 191