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, γs and γi, conditional coincidence, µ i|s , and pair coupling, γc, 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 nm laser creating photon pairs at 810 nm and 1550 nm. Numerical calculations lead to coupling efficiencies above 93% at optimal focusing, which is found by the geometrical relation L/zR to be ≈ 1 to 2 for the pump mode and ≈ 2 to 3 for the fiber-modes, where L is the crystal length and zR is the Rayleigh-range of the mode-profile. These results are independent on L. By showing that the single-mode bandwidth decreases ∝ 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 ∝ √ L, and so, assuming correct focusing, we can only see advantages using long crystals.

Optimal collinear Gaussian beams for spontaneous parametric down-conversion

“…This makes sense, since SHG with a monochromatic Gaussian input beam produces a monochromatic Gaussian second harmonic field; optimization of SHG then amounts to finding the parameters of the Gaussian modes that maximize the spatial overlap, which is precisely what was done in section III. More recently, Ljunggren and Tengner [14] have performed numerical studies yielding the optimizing conditions 2ξ p = 1.7, 2ξ s = 2ξ i = 2.3, and the prediction that the optimized joint probability goes as √ L (rather than being independent of L as claimed here). These discrepencies may be due to two key differences in approach: In [14], optimal focusing conditions are obtained for Φ = 0, whereas here and in [19] the condition Φ = −1.04π is found to yield a slightly higher spectral density.…”

“…More recently, Ljunggren and Tengner [14] have performed numerical studies yielding the optimizing conditions 2ξ p = 1.7, 2ξ s = 2ξ i = 2.3, and the prediction that the optimized joint probability goes as √ L (rather than being independent of L as claimed here). These discrepencies may be due to two key differences in approach: In [14], optimal focusing conditions are obtained for Φ = 0, whereas here and in [19] the condition Φ = −1.04π is found to yield a slightly higher spectral density. Additionally, [14] employs a plane-wave approach in which the diffraction of the pump mode is effectively ignored; this approximation is also made in [13], [11] and [15].…”

“…To the extent that the parameter C (see (14)) is negligibly small, the joint collection probability, heralding ratios, and spectral purity are determined by the dimensionless ratios L/k j w 2 j and ∆ω p L/c. Changing the crystal length has no effect on these three quantities if the pump duration and confocal ranges are also scaled by the same factor.…”

“…As brightness has improved, interest has progressed from simply making SPDC sources brighter to controlling other properties of the emitted photons, such as the spectral entanglement in each photon pair [5][6][7][8]. While several papers in recent years [9][10][11][12][13][14][15][16] have addressed the question of how to focus the pump and/or collection optics optimally, some important questions remain. Since different studies have invoked slightly different assumptions and optimized slightly different measures, it is not clear how pump and/or collection focusing simultaneously affects all the quantities that are generally of interest, and what trade-offs (if any) exist between these quantities.…”

Bell violation using entangled photons without the fair-sampling assumption

“…We couple the 810 nm photons into an optical fiber (SMF-28), which guides the photons to the sensitive area of the detectors. To achieve high coupling efficiency in both arms, we optimized the focusing of the pump laser and the fiber couplers (24,25).…”