  We present a novel fiber-optic component, a "tapered multi-core fiber (MCF)",
designed for integration into ultra-miniaturized endoscopes for minimally
invasive two-photon point-scanning imaging and to address the power delivery
issue that has faced MCF based lensless endoscopes. With it we achieve
experimentally a factor 6.0 increase in two-photon signal yield while keeping
the ability to point-scan by the memory effect, and a factor 8.9 sacrificing
the memory effect. To reach this optimal design we first develop and validate a
fast numerical model capable of predicting the essential properties of an
arbitrarily tapered MCF from its structural parameters. We then use this model
to identify the tapered MCF design parameters that result in a chosen set of
target properties (point-spread function, delivered power, presence or absence
of memory effect). We fabricate the identified target designs by stack-and-draw
and post-processing on a CO$_{2}$ laser-based glass processing and splicing
system. Finally we demonstrate the performance gain of the fabricated tapered
MCFs in two-photon imaging when used in a lensless endoscope system. Our
results show that tailoring of the taper profile brings new degrees of freedom
that can be efficiently exploited for lensless endoscopes.
