  Optically localizing a single quasi-monochromatic source to sub-diffractive
precisions entails, in the photon-counting limit, a minimum photon cost that
scales as the squared ratio of the width, $w$, of the optical system's
point-spread function (PSF) and the sought localization precision, $d$, i.e.,
as $\alpha(w/d)^2$. For sources with a finite emission-frequency spectrum,
while the inverse quadratic scaling is expected to remain unchanged, the
coefficient $\alpha$ must increase due to a degrading fidelity of localization
as the imaging bandwidth increases and PSF undergoes a frequency-dependent
widening. We specifically address how rapidly $\alpha$ must increase with
increasing width of a flat-top spectral profile of emission of a point source
being localized in two dimensions by an imager with a clear circular aperture
by calculating quantum Fisher information (QFI), whose inverse yields the
lowest possible unbiased-estimation variance of source-localization error. The
novel use of prolate spheroidal wave functions as a basis for obtaining a
solution of the eigenvalue problem of the single-photon density operator needed
for the QFI calculation helps us develop the notion of an effective
dimensionality of the continuous-state problem in terms of the associated
space-bandwidth parameter. We subsequently extend our considerations of QFI to
treat the finite-bandwidth pair superresolution problem in two dimensions,
obtaining similar results. We also consider generalizations to emission power
spectra of arbitrary profiles.
