14 May 2018 An analytic method for spectrum recovery from wedge or staircase spectrometers
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Abstract
Wedge or staircase micro-optics have become important components for building miniature optical spectrometers. These devices create spectral discrimination through interference between beams resulting from reflections at the surfaces of the optic. The literature has examples of low reflectance wedge spectrometer system where the Fourier transform is used to recover the spectrum (with no inherent bandwidth limit), and high-reflectance, band-limited simplex spectrometers where no data processing is required. Instruments in the first category tend to be for the thermal infrared range, and instruments in the second category are more often encountered in the visible band. This second category includes linear variable filters and discrete etalon staircases. Though in practice, the signal treatment for these two types of spectrometers is radically different, the underlying interference mechanism is identical. It follows, that a single signal processing algorithm must exist which correctly treats the two types of signals. We present a mathematical description of the signal model for such spectrometers. We show that in the case of spectrally uniform reflectance, the signal has a specific relationship to the spectrum’s Fourier transform. We cast the spectral recovery problem as a matrix inversion, and derive formulas for calculating the solution matrix. The solution matrix is shown to yield the exact spectrum when applied to modeled wedge spectrometer signals in both low and high reflectance cases.
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Michael K. Yetzbacher, Michael K. Yetzbacher, Christopher W. Miller, Christopher W. Miller, Michael J. DePrenger, Michael J. DePrenger, "An analytic method for spectrum recovery from wedge or staircase spectrometers ", Proc. SPIE 10657, Next-Generation Spectroscopic Technologies XI, 106570F (14 May 2018); doi: 10.1117/12.2303980; https://doi.org/10.1117/12.2303980
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