Highly multi-mode media convert a coherent input field’s spatial profile into a granular speckle patterns. These speckle patterns are heavily dependent upon properties of the multi-mode medium and the input light field. Examining these speckle patterns enables extraction of these parameters; numerous spectrometers have been demonstrated by analyzing speckle. This is germane to precision measurement of optical wavelength, which is crucial in many applications including: laser spectroscopy, optical sensing, and laser locking. To the best of our knowledge we have demonstrated, for the first time, a transmission matrix method (TMM) speckle spectrometer capable of resolving the hyperfine structure of rubidium at 780nm. We compare this with other TMM speckle spectrometers in literature as well as discussing the advantages and disadvantages of the transmission matrix technique.
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