Designing a non-scanning imaging spectrometer

Abstract

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 f(λ) ≈ nₓΔₓ fₓ+n(y)Δ(y)f(y) where f(λ) is a Fourier decomposition of the wavelength spectrum across the CTIS spectral bandwidth, fₓ and f(y) are the horizontal and vertical spatial frequencies, nₓ and n(y) are the diffractive-order numbers as would be obtained by crossed diffraction gratings, and Δₓ 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 (nₓ and n(y) can both be non-zero) and anamorphic dispersion (Δₓ ≠ Δ(y)).