Imaging Spectroscopy Tutorial 

Imaging Spectrometers

Imaging spectrometers acquire a data cube consisting of two spatial and one spectral dimension. There are a variety to do this, depending on whether spectra are dispersed on 0, 1, or 2 dimensions. The availability of focal plane detectors arrays enables highly efficient acquisition of spectrally and spatially resolved data.

Examples of imaging spectrometers are

For example, a long slit grating spectrometer which disperses light in 1 spatial dimension and provides imaging data in an orthogonal direction. By scanning of a succession of 1-d slices of the scene a 2-d image may be reconstructioned. Other variants of the DS, are the intral field unit (IFU) and the multi-object spectrometer (MOS), which use different strategies to project the slit on the sky. The IFU slices the slit into short sections, and stacks the slices side by side to provide a contiguous view of thes sky. The MOS either uses fibers or a focal plane mask of slitlets. This could be a filter wheel or Fabry-Perot. An imaging TF acquires a full 2-d image each frame, with successive frames viewed through successive filters. In this case the light is not dispersed, but only a narrow bandpass is transmitted to the detector array. An IFTS acquires a full 2-d image per frame, with successive frames associated with different positions of a moving mirror in an interferometer. The raw data cube for an IFTS consists of the interferogram in 1-d (time) and the 2-d scene in the 2 orthogonal directions of the detector. Fourier transformation of the interferograms for each pixel produces the spectral dimension of the data cube.
The number of spectral channels practical with an imaging DS is limited by the number of pixels of the focal plane image in one dimension. In practice the number of spectral channels is at most a few thousand. Imaging TF systems are limited in practice to a few tens to hundreds of discrete filter elements.

Basic Principles of IFTS

Fourier transform spectrometers are based on the Michelson interferometer, with one fixed mirror and one moving mirror. The light transmitted through the interferometer is measured as a function of the displacement of the moving mirror from the zero phase difference (ZPD) position. The Fourier transform of this interferogram yields the spectrum. An imaging spectrometer is obtained by viewing the scence through the interferometer with a camera, and constructing the Fourier transform of the variations in light intensity at each pixel as a function of the position of the moving mirror.

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 tex2html_wrap_inline32 , and the wavelength, tex2html_wrap_inline34 . For monochromatic light of wavenumber tex2html_wrap_inline36 , the intensity of light at the image point y' relative to the intensity emerging from the object point y is proportional to tex2html_wrap_inline42 . The phase difference tex2html_wrap_inline44 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 tex2html_wrap_inline48 , 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 tex2html_wrap_inline68 , 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 tex2html_wrap_inline32 and spread tex2html_wrap_inline84 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.

Reference: Bennett, C. L., Carter, M. R., Fields, D. J. & Hernandez, J. 1993,
Proc. SPIE, 1937, 191