Abstract
Using Fisher information and the Cramér-Rao lower bound, we analyse fundamental precision limits in the determination of spectral parameters in inelastic optical scattering. General analytic formulae are derived which account for the instrument response functions of the dispersive element and relay optics found in practical Raman and Brillouin spectrometers. Limiting cases of dispersion and diffraction limited spectrometers, corresponding to measurement of Lorentzian and Voigt lineshapes respectively, are discussed in detail allowing optimal configurations to be identified. Effects of defocus, spherical aberration, detector pixelation and a finite detector size are also considered.
Highlights
Using Fisher information and the Cramér-Rao lower bound, we analyse fundamental precision limits in the determination of spectral parameters in inelastic optical scattering
In the case where read out noise is not the dominate noise source, a Gaussian probability density function (PDF) can still provide a good approximation to a Poisson PDF if the mean intensity is large enough
In a similar vein to above we can determine the Fisher information matrix (FIM) for an infinite-extent, finely pixelated detector, instead of assuming the intensity distribution on the detector is limited by the response function of the dispersive element, we can instead consider the case in which the PSF, hopt(x), of the relaying optics dominates
Summary
Using Fisher information and the Cramér-Rao lower bound, we analyse fundamental precision limits in the determination of spectral parameters in inelastic optical scattering. Interactions between either acoustic or optical phonons in a material and an incident photon can give rise to inelastic scattering, known as Brillouin or Raman scattering respectively1,2 Such processes provide a means by which to probe the vibrational, micro-mechanical and compositional properties of samples. Systematic quantitative analysis of achievable precision in inelastic optical spectroscopy is, hitherto lacking and will form the focus of this article Evaluation of such limits is of importance in terms of aiding system design, in scenarios with limited photon budgets, but can enable benchmarking of data processing algorithms and analysis protocols. We proceed to outline the information theoretic precision limit, given by the Cramér-Rao lower bound, as applied to the problem of extracting spectral parameters in inelastic spectroscopy We apply these results to a number of limiting cases and numerical examples
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