Since its inception in the early 1960s [1], a nonlinear optical diffraction [2] technique known as Quasi-phase-matching (QPM) was used to mitigate the dispersion-born phase mismatch in optical processes involving beams with different frequencies. QPM is relevant to many areas of photonic research and applications as it allows among other things: efficient frequency conversion, beam and pulse shaping, multi-harmonic generation, high-harmonic-generation, all-optical processing and generation of entangled photons. Since QPM was first demonstrated using a one-dimensional periodic modulation it evolved to include quasi-periodic modulations, random modulations, two-dimensional structures and also non-uniform structures. In all of its various forms QPM always used a spatial-modulation (usually of the nonlinear electric susceptibility) to account for the momentum imbalance that dispersion imposes on the interacting photons.

Here we present a fundamental extension of quasi phase matching where spatiotemporal nonlinear optical diffraction allows for both energy and momentum imbalance to be corrected-for. In the regular, traditional, QPM process, the spatial modulation breaks the otherwise continuous spatial translational symmetry, experienced by the photons, which leads to the requirement of exact momentum conservation. Adding a temporal modulation allows to replace the usual requirement for exact conservation of energy with a more flexible condition which depends on the exact nature of the temporal modulation used. In this case the phase mismatch caused by dispersion can be distributed between the momentum and energy four-momentum components of the interacting photons. This spatiotemporal QPM establishes a framework for designing and analyzing phase-matching schemes based on dynamic modulations. The most notable example for using such a dynamic modulation is the all-optical QPM scheme which uses counter propagating light pulses to establish a moving modulation of the nonlinear interaction [3]. This example also proves that spatiotemporal QPM is a feasible concept. More importantly, spatiotemporal QPM would allow unique schemes which were impossible with the regular spatial QPM. For example walk-off losses can be avoided when phase matching several different processes by the use of a nonlinear spatiotemporal photonic crystal. Parameters changing temporally (such as frequency chirp) could be addressed more efficiently than they do today [4]. And finally, being able to distribute the effect of dispersion between the spatial and temporal components would allow further flexibility in the design of new QPM schemes.  

 References

[1] J. A. Armstrong et al.,  Phys. Rev. 127, 1918–1939 (1962).

[2] I. Freund, Phys. Rev. Lett. 21, 1404 (1968).

[3] X. Zhang et al., Nature Physics 3, 270–275 (2007).

[4] M. Arbore, O. Marco, and  M. Fejer, Opt. Lett. 22, 865 (1997).