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Recent work considered the ultimate (quantum) limit of the precision ofestimating the distance between two point objects. It was shown that theperformance gap between the quantum limit and that of ideal continuumimage-plane direct detection is the largest for highly sub-Rayleigh separationof the objects, and that a pre-detection mode sorting could attain the quantumlimit. Here we extend this to a more general problem of estimating the lengthof an incoherently radiating extended (line) object. We find, as expected bythe Rayleigh criterion, the Fisher information (FI) per integrated photonvanishes in the limit of small length for ideal image plane direct detection.Conversely, for a Hermite-Gaussian (HG) pre-detection mode sorter, thisnormalized FI does not decrease with decreasing object length, similar to thetwo point object case. However, unlike in the two-object problem, the FI perphoton of both detection strategies gradually decreases as the object lengthgreatly exceeds the Rayleigh limit, due to the relative inefficiency ofinformation provided by photons emanating from near the center of the objectabout its length. We evaluate the quantum Fisher information per unitintegrated photons and find that the HG mode sorter exactly achieves this limitat all values of the object length. Further, a simple binary mode sortermaintains the advantage of the full mode sorter at highly sub-Rayleigh length.In addition to this FI analysis, we quantify improvement in terms of the actualmean squared error of the length estimate. Finally, we consider the effect ofimperfect mode sorting, and show that the performance improvement over directdetection is robust over a range of sub-Rayleigh lengths.

Attaining the quantum limit of super resolution in imaging an object's length via pre-detection spatial mode sorting