  Photonic integrated optical components, notably straight waveguides, serve as
pivotal elements for on-chip generation and manipulation of quantum states of
light. In this work, we focus on optimizing waveguides based on lithium niobate
on insulator (LNOI) to generate photon pairs at telecom wavelength using
spontaneous parametric down-conversion (SPDC). Specifically, we investigate
lateral leakage for all possible SPDC processes involving type 0, type I and
type II phase matching conditions in an X-cut lithium niobate waveguide and
provide a recipe to avoid leakage loss for the interacting photons.
Furthermore, focusing on type II phase matching, we engineer the waveguide in
the single mode regime such that it also satisfies group index matching for
generating spectrally pure single photons with high purity (99.33%). We also
address fabrication imperfections of the optimized design and found that the
spectral purity of the generated photons is robust to fabrication errors. This
work serves as a tutorial for the appropriate selection of morphological
parameters to obtain lossless, single mode LNOI waveguides for building linear
optical circuits and photon pair generation at telecom wavelengths using
desired phase-matching conditions.
