Ultra-high precision speckle spectrometer enabling radio-frequency scale resolution of atomic spectra
Laser speckle, the granular intensity pattern arising from random optical interference, provides a high-dimensional encoding of spectral information that can be exploited for precision metrology. Speckle-based spectrometers have advanced rapidly owing to their compact footprint, mechanical robustness and alignment agnostic nature, yet their spectral resolution has remained limited to the picometre scale. In this work, we break this limit by employing an integrating sphere as a multiply scattering cavity with access to a high range of path lengths to enhance spectral sensitivity. At 780$\,$nm, the resulting device achieves a resolution of 6$\,$fm, corresponding to a resolving power of $1.3\times10^8$, representing an approximately 80-fold improvement over previous implementations. This ultra-high resolution enables clear discrimination of laser sidebands generated by an electro-optical modulator, with extracted sideband powers agreeing with expected values to within 1%. It further permits the first direct speckle-based measurement of the hyperfine structure of the $\text{D}_{2}$ transition in $^{85}\text{Rb}$, with transmission spectra differing by no more than 3.6% from independent wavemeter-referenced measurements. These results establish speckle as a new platform for ultra-high precision spectroscopy, radio-frequency spectrometry, and microwave photonics.