= Solution
An <integral field spectrograph> obtains spectra throughout a two-dimensional field instead of along one slit alone. Its reduced <spectral data cube> is $I(x,y,\lambda)$: two coordinates locate a spatial sampling element and the third labels <wavelength>. One slice at fixed <wavelength> is an image; one column at fixed spatial position is an <optical spectrum>. The third axis is spectral, not a third spatial direction.
Six ways to obtain such a cube illustrate the distinction between field reformatting and scanning:
* A <lenslet array> divides the image into spatial elements. The <spectrograph> disperses each lenslet’s output into a microspectrum, with spacing and orientation chosen to avoid overlap.
* A <lenslet array> feeding <optical fibers> provides contiguous sampling; the fiber outputs are rearranged into a pseudo-slit for a conventional <spectrograph>.
* A bare <optical fiber> bundle samples the image and also reformats it into a pseudo-slit. Its packing fraction and coupling can leave spatial gaps or transmission losses.
* An <image slicer> divides the image into strips, then rearranges them end to end. Dispersion preserves position along each strip, and the reconstruction restores the second spatial coordinate.
* A scanning <Fabry–Pérot interferometer> or tunable narrow-band filter records a sequence of nearly monochromatic images, stepping the transmitted band to assemble the cube.
* Imaging <Fourier transform spectroscopy> records an interferogram at every spatial pixel while scanning <optical path length> difference. Transforming each interferogram supplies the spectral coordinate.
The first four are simultaneous spatially multiplexed grating arrangements. The last two deliver equivalent cube coordinates by scanning; they are imaging spectrometers rather than simultaneous grating integral-field units, and variability during the scan can corrupt the cube. Detector packing, sampling, calibration and throughput determine the practical tradeoffs.
\b[A data cube contains one spectrum per spatial element: two sky coordinates plus wavelength.]
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