Designing a non-scanning imaging spectrometer

A non-scanning imaging spectrometer simultaneously captures spatial and spectral information via multiple diffractive orders. Optics image a color scene in a field stop. A collimating lens converts the scene's spatial information into propagation angles. A diffractive disperser multiplexes the scene's spectral information into the propagation angles. A lens focused at infinity images multiple diffractive orders onto a large sensor array, which cannot distinguish the wavelength of incident light within the spectral bandpass of the instrument. The pixels of the sensor array collapse the two-spatial, one-spectral dimensions into a discrete, two-dimensional array. This collapsing of three dimensions into two is a mathematical projection. Computed tomography uses projections to reconstruct a three-dimensional object. Hence, this non-scanning imaging spectrometer has become known as the Computed-Tomography Imaging Spectrometer, or CTIS.; The results imply nominal spatial and spectral resolution limits. When each projection is considered separately, the Nyquist spatial-sampling criterion provides a resolution limit. The limit cannot be achieved for an arbitrary scene. The highest spectral resolution can be obtained only if the highest spatial frequency is present. The formula that defines what each diffractive order measures is fl≈nxDx fx+nyDyfy where f_λ is a Fourier decomposition of the wavelength spectrum across the CTIS spectral bandwidth, f _x and f_y are the horizontal and vertical spatial frequencies, n_x and n_y are the diffractive-order numbers as would be obtained by crossed diffraction gratings, and Δ_x and Δ_y are established by the optical design. Derived from a simple model of scalar diffraction, the formula is shown to be consistent with CTIS calibrations using a technique from computed tomography known as the Fourier-crosstalk matrix. The formula extends the definition of what CTIS projections measure to include cross-orders (n_x and n_y can both be non-zero) and anamorphic dispersion (Δ_x ≠ Δ _y).