  The rapidly developing quantum communication technology requires
deterministic quantum emitters that can generate single photons and entangled
photon pairs in the third telecom window, in order to be compatible with
existing optical fiber networks and on-chip silicon photonic processors.
InAs/InP quantum dots (QDs) are among the leading candidates for this purpose,
due to their high emission efficiency in the required spectral range. However,
fabricating versatile InAs/InP QD-based quantum emitters is challenging,
especially as these QDs typically have asymmetric profiles in the growth plane,
resulting in a substantial bright-exciton fine structure splitting (FSS). This
hinders the generation of entangled photon pairs and thus, compromises the
versatility of InAs/InP QDs. We overcome this by implementing droplet epitaxy
(DE) synthesis of low surface density (2.8$\times$10$^8$ cm$^{-2}$) InAs QDs on
an (001)-oriented InP substrate. The resulting QDs are located in etched pits,
have concave bases, and most importantly, have symmetric in-plane profiles. We
provide an analytical model to explain the kinetics of pit formation and QD
base shape modification. Our theoretical calculations of electronic states
reveal the properties of neutral and charged excitons and biexcitons confined
in such QDs, which agree with the optical investigations of individual QDs. The
optical response of QD ensembles suggests that FSS may indeed be negligible, as
reflected in the vanishing degree of linear polarization. However, single QD
spectrum gathered from an etched mesa shows moderate FSS of (50$\pm$5) $\mu$eV
that we link to destructive changes made in the QD environment during the
post-growth processing. Finally, we show that the studied DE QDs provide a
close-to-ideal single-photon emission purity of (92.5$\pm$7.5) $\%$ in the
third telecom window.
