  Miniaturization of force probes into nanomechanical oscillators enables
ultrasensitive investigations of forces on dimensions smaller than their
characteristic length scale. Meanwhile it also unravels the force field
vectorial character and how its topology impacts the measurement. Here we
expose an ultrasensitive method to image 2D vectorial force fields by
optomechanically following the bidimensional Brownian motion of a singly
clamped nanowire. This novel approach relies on angular and spectral tomography
of its quasi frequency-degenerated transverse mechanical polarizations:
immersing the nanoresonator in a vectorial force field does not only shift its
eigenfrequencies but also rotate eigenmodes orientation as a nano-compass. This
universal method is employed to map a tunable electrostatic force field whose
spatial gradients can even take precedence over the intrinsic nanowire
properties. Enabling vectorial force fields imaging with demonstrated
sensitivities of attonewton variations over the nanoprobe Brownian trajectory
will have strong impact on scientific exploration at the nanoscale.
