Spectral Information Content of Diffraction-Limited Images

Abstract

High-resolution optical imaging is fundamentally limited by diffraction, which imprints wavelength-dependent structure onto the point spread function (PSF) of the optical system. This intrinsic chromaticity implies that in addition to spatial structure, images encode spectral information. We develop the theory of this spectral information content and provide strategies for its optimal extraction with calculations of Fisher information. We show that polychromatic PSF formation can be expressed as a convolution in logarithmic coordinates and define the corresponding spectral transfer function. We derive this function for the Airy disk and find algebraic decay in spectral log-frequency, implying the existence of arbitrarily fine spectral structure in the ideal, noiseless PSF. We use a differentiable forward model to numerically construct the spectral information eigenbasis and quantify the spectroscopic capability of diffraction-limited imaging across a range of noise and detector systematics. We demonstrate injection and recovery of the constrained spectral modes in a realistic simulated example and compare these results with previously reported methods. Our information-theoretic framework provides a generalisable method for spectral characterisation of PSFs spanning a variety of imaging contexts.