Abstract
Precisely measuring molecular orientation is key to understanding how molecules organize and interact in soft matter, but the maximum theoretical limit of measurement precision has yet to be quantified. We use quantum estimation theory and Fisher information (QFI) to derive a fundamental bound on the precision of estimating the orientations of rotationally fixed molecules. While direct imaging of the microscope pupil achieves the quantum bound, it is not compatible with wide-field imaging, so we propose an interferometric imaging system that also achieves QFI-limited measurement precision. Extending our analysis to rotationally diffusing molecules, we derive conditions that enable a subset of second-order dipole orientation moments to be measured with quantum-limited precision. Interestingly, we find that no existing techniques can measure all second moments simultaneously with QFI-limited precision; there exists a fundamental trade-off between precisely measuring the mean orientation of a molecule versus its wobble. This theoretical analysis provides crucial insight for optimizing the design of orientation-sensitive imaging systems.
Highlights
Since the first observation of single molecules [1], scientists and engineers have worked tirelessly to quantify precisely their positions [2,3,4] and orientations [5,6,7,8,9] to probe dynamic processes within soft matter at the nanoscale
We derive a fundamental bound for estimating the orientation of rotationally fixed molecules that applies to all measurement techniques
Estimation performance can vary dramatically depending on how the background photons interact with the signal photons and the parameters to be estimated, and exploring these effects for typical single-molecule imaging conditions remains the object of future study
Summary
Since the first observation of single molecules [1], scientists and engineers have worked tirelessly to quantify precisely their positions [2,3,4] and orientations [5,6,7,8,9] to probe dynamic processes within soft matter at the nanoscale. I.e., Fisher information (FI) and the associated CramérRao bound (CRB) [15], allows us to calculate conveniently the best possible precision of unbiased measurements of a few parameters. Since quantum noise manifests itself as shot noise in incoherent optical imaging systems, the quantum Cramér-Rao bound (QCRB) sets a fundamental limit on the best possible variance of measuring any parameter of interest. This approach provides insight into how one may design an instrument to saturate the quantum bound, thereby achieving a truly optimal imaging system [19,20].
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