Optimal focusing for maximal collection of entangled narrow-band photon pairs into single-mode fibers

Abstract

We present a theoretical and experimental investigation of the emission characteristics and the flux of photon pairs generated by spontaneous parametric downconversion in quasi-phase matched bulk crystals for the use in quantum communication sources. We show that, by careful design, one can attain well defined modes close to the fundamental mode of optical fibers and obtain high coupling efficiencies also for bulk crystals, these being more easily aligned than crystal waveguides. We distinguish between singles coupling, ${\ensuremath{\gamma}}_{\mathrm{s}}$ and ${\ensuremath{\gamma}}_{\mathrm{i}}$, conditional coincidence, ${\ensuremath{\mu}}_{\mathrm{i}\ensuremath{\mid}\mathrm{s}}$, and pair coupling, ${\ensuremath{\gamma}}_{\mathrm{c}}$, and show how each of these parameters can be maximized by varying the focusing of the pump mode and the fiber-matched modes using standard optical elements. Specifically we analyze a periodically poled KTP-crystal pumped by a $532\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ laser creating photon pairs at $810\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and $1550\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Numerical calculations lead to coupling efficiencies above 93% at optimal focusing, which is found by the geometrical relation $L∕{z}_{R}$ to be $\ensuremath{\approx}1$ to 2 for the pump mode and $\ensuremath{\approx}2$ to 3 for the fiber-modes, where $L$ is the crystal length and ${z}_{R}$ is the Rayleigh-range of the mode-profile. These results are independent on $L$. By showing that the single-mode bandwidth decreases $\ensuremath{\propto}1∕L$, we can therefore design the source to produce and couple narrow bandwidth photon pairs well into the fibers. Smaller bandwidth means both less chromatic dispersion for long propagation distances in fibers, and that telecom Bragg gratings can be utilized to compensate for broadened photon packets---a vital problem for time-multiplexed qubits. Longer crystals also yield an increase in fiber photon flux $\ensuremath{\propto}\sqrt{L}$, and so, assuming correct focusing, we can only see advantages using long crystals.