Measurement of variations in gas refractive index with $10^{-9}$ resolution using laser speckle

Highly-resolved determination of refractive index is vital in fields ranging from biosensing through to laser range-finding. Laser speckle is known to be a sensitive probe of the properties of the light and the environment, but to date speckle-based refractive index measurements have been restricted to $10^{-6}$ resolution. In this work we identify a strategy to optimise the sensitivity of speckle to refractive index changes, namely by maximising the width of the distribution of optical path lengths in the medium. We show that this can be realised experimentally by encapsulating the medium of interest within an integrating sphere. We demonstrate that variations of the refractive index of air as small as $4.5\times10^{-9}$ can be resolved with an uncertainty of $7\times10^{-10}$. This is an improvement of three orders of magnitude when compared to previous speckle-based methods.