  This paper describes a new method for optical readout of Time Projection
Chambers (TPCs), based on the Linearly Graded Silicon Photomultiplier
(LG-SiPM). This is a single photon-sensitive detector with excellent timing and
2D position resolution developed at Fondazione Bruno Kessler, Trento (FBK). The
LG-SiPM produces time-varying voltage signals that are used to reconstruct the
3D position and energy of ionisation tracks generated inside the TPC. The TPC
used in this work contained room-temperature CF$_4$ gas at a pressure of 100
mbar, with two THGEMs to produce secondary scintillation light. A collimated
$^{241}$Am source (Q$_\alpha$ = 5.486 MeV) was used to produce the ionisation
tracks. The successful reconstruction of these tracks is demonstrated, and the
consistency of the methodology characterised through varying the geometry of
the tracks within the TPC. Energy reconstruction and deposition studies are
also described, demonstrating the feasibility of the LG-SiPM as a potential
option for optical TPC readout.
