  Spontaneous parametric down-conversion (SPDC) is a widely used process to
prepare entangled photon pairs. In SPDC, a second-order nonlinear crystal is
pumped by a coherent laser beam to generate photon pairs. The photon pairs are
usually detected by single-mode fibers (SMF), where only photons in a Gaussian
mode can be collected. The collection modes possess typical Gaussian
parameters, namely a beam waist and a focal plane position. The collection
efficiency of photons highly depends on the choice of both parameters. The
exact focal plane position of the pump beam relative to those of the detection
modes is difficult to determine in a real experiment. Usually, theoretical and
experimental studies assume that the focal plane positions of the pump and the
generated beams are positioned in the center of the crystal. The displacement
of beam focal planes can lead to deviations from expected results and the
coupling efficiency into SMF can increase or decrease. In this work, we
consider variable positions of focal planes and investigate how shifts of these
focal planes influence the spatial and temporal properties of photon pairs. We
present SPDC arrangements, in which the knowledge of the exact position of the
focal planes is essential, as well as scenarios, where focal plane
displacements do not contribute significantly to experimental outcomes. These
findings are of particular interest for achieving higher efficiency in SPDC
experiments.
