Information bounds in determining the 3D orientation of a single emitter or scatterer using point-detector-based division-of-amplitude polarimetry

Determining the 3D orientation of a single molecule or particle, encoded in its polar and azimuthal angles, is of interest for a variety of fields, being relevant to a range of questions in elementary chemical reactivity, biomolecular motors, and nanorheology. A popular experimental method, known as division-of-amplitude polarimetry, for determining the real-time orientation of a single particle is to split the emitted/scattered light into multiple polarizations and to measure the light intensity using point detectors at these polarizations during a time interval Δt. Here, we derive the Cramér-Rao lower bounds for this method from the perspective of information theory in the cases of utilizing a chromophore or a scattering particle as a 3D orientation probe. Such Cramér-Rao lower bounds are new for using this experimental method to measure the full 3D orientation in both the scattering case and the fluorescence case. These results show that, for a scatterer, the information content of one photon is 1.16 deg⁻² in the polar and 58.71 deg⁻² in the azimuthal angles, respectively. For a chromophore, the information content of one photon is 2.54 deg⁻² in the polar and 80.29 deg⁻² in the azimuthal angles. In addition, the Cramér-Rao lower bound scales with the square root of the total signal photons. To determine orientation to an uncertainty of one degree requires 7.40 × 10⁴ and 2.34 × 10³ photons for the polar and the azimuthal angles, respectively, for fluorescence, whereas it takes 1.62 × 10⁵ and 3.20 × 10³ photons for scattering. This work provides experimentalists new guidelines by which future experiments can be designed and interpreted.