From Ewald sphere to Ewald shell in nonlinear optics

Stemming from the X-ray diffraction, the Ewald sphere is a convenient method to depict the Bragg diffraction in a crystal, where the diffraction resonance arises when a reciprocal lattice point is located on the Ewald sphere1. The Ewald construction was also employed for the photonic crystals to study the light diffraction properties, which reveals the structural symmetry of artificial crystals or enhances light extraction from the light emitting diodes2,3. The Ewald sphere may be extended to the nonlinear optics as well. In 1998, Berger proposed the concept of nonlinear photonic crystal (NPC)4, which owns a homogeneous dielectric constant and a modulated nonlinear coefficient. For 1D NPC such as the domain-inverted ferroelectric crystals, QPM frequency conversion usually occurs in the collinear geometry5,6,7 with exceptional early works on the non-collinear cases, e.g., I. Freund’s study of nonlinear diffraction8. For the 2D counterpart, however, non-collinear QPM (or nonlinear diffraction) and multiple QPM processes become popular because of more reciprocal lattice points provided4,9,10,11. To describe the non-collinear effect geometrically, nonlinear Ewald sphere construction was proposed4, which indicates that, if a point of reciprocal lattice intersects the nonlinear Ewald sphere, a QPM resonance will be resulted. The method is also useful for studying some other nonlinear effects12,13,14,15,16, such as the scattering-assisted conical second-harmonic (SH) generation, the nonlinear Cherenkov radiation, etc. Here, we extend the concept of nonlinear Ewald sphere to the nonlinear Ewald shell. As we know, the non-collinear QPM effect relies on the dimensions of Ewald sphere and reciprocal lattice. If the period of NPC is rather small and the reciprocal lattice points distribute sparsely, there will be few or discrete QPM resonances. But given a dense distribution of reciprocal points, more QPM processes may come into being and thus present new effects. Following this idea, a collective envelope effect because of multiple QPM resonances has been observed and analysed with the Ewald shell. An enhanced SH due to the simultaneous participation of multiple reciprocal vectors has also been suggested. Moreover, the dynamic evolution of these effects was experimentally studied, which matches well with the Ewald shell model.