Within the framework of quantum mechanics, a unique particle wave packet exists in the form of the Airy function. Its counterintuitive properties are revealed as it propagates in time or space: the quantum probability wave packet preserves its shape despite dispersion or diffraction and propagates along a parabolic caustic trajectory, even though no force is applied. This does not contradict Newton's laws of motion, because the wave packet centroid propagates along a straight line. Nearly 30 years later, this wave packet, known as an accelerating Airy beam, was realized in the optical domain; later it was generalized to an orthogonal and complete family of beams that propagate along parabolic trajectories, as well as to beams that propagate along arbitrary convex trajectories. Here we report the experimental generation and observation of the Airy beams of free electrons. These electron Airy beams were generated by diffraction of electrons through a nanoscale hologram, which imprinted on the electrons' wavefunction a cubic phase modulation in the transverse plane. The highest-intensity lobes of the generated beams indeed followed parabolic trajectories. We directly observed a non-spreading electron wavefunction that self-heals, restoring its original shape after passing an obstacle. This holographic generation of electron Airy beams opens up new avenues for steering electronic wave packets like their photonic counterparts, because the wave packets can be imprinted with arbitrary shapes or trajectories.

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Introducing structure into photon pair generation via spontaneous parametric down‐conversion (SPDC) is shown to be useful for controlling the output state and exploiting new degrees of freedom for quantum technologies. This paper presents a new method for simulating first‐ and second‐order correlations of the down‐converted photons in the presence of structured pump beams and shaped nonlinear photonic crystals. This method is nonperturbative, and thus accounts for high‐order effects, and can be made very efficient using parallel computing. Experimental results of photodetection and coincidence rates in complex spatial configurations are recovered quantitatively by this method. These include SPDC in 2D nonlinear photonic crystals, as well as with structured light beams such as Laguerre Gaussian and Hermite Gaussian beams. This simulation method reveals conservation rules for the down‐converted signal and idler beams that depend on the nonlinear crystal modulation pattern and the pump shape. This scheme can facilitate the design of nonlinear crystals and pumping conditions for generating non‐classical light with pre‐defined properties.

We report analytical solutions for spatial solitons supported by layers of a quadratically nonlinear (χ (2) ) material embedded into a linear planar waveguide. A full set of symmetric, asymmetric, and antisymmetric modes pinned to a symmetric pair of the nonlinear layers is obtained. The solutions describe a bifurcation of the subcritical type, which accounts for the transition from the symmetric to asymmetric modes. The antisymmetric states (which do not undergo the bifurcation) are completely stable (the stability of the solitons pinned to the embedded layers is tested by means of numerical simulations). Exact solutions are also found for nonlinear layers embedded into a nonlinear waveguide, including the case when the uniform and localized χ (2) nonlinearities have opposite signs (competing nonlinearities). For the layers embedded into the nonlinear medium, stability properties are explained by comparison to the respective cascading limit.+∞ −∞ |u(x)| 2 dx ≡ 1. In particular, the formal application of the Vakhitov-Kolokolov (VK) criterion, dP/dµ < 0, which is a necessary stability condition for solitons in self-focusing nonlinear media [8], predicts neutral stability of solutions (3). In fact, all these degenerate solitons are unstable, collapsing into a singularity or decaying, as illustrated by another analytical solution to Eq. (1), which explicitly describes the onset of the collapse at z → −0 [9]:Here x 0 > 0 is an arbitrary real constant, and z is negative. The same solution (4) with x 0 < 0 describes decaying solitons at z > 0 [9]. The power of this solution is also P = 1, irrespective of the value of x 0 . The solitons may be stabilized if a linear periodic potential is added to Eq. (1) [9]. The stability is also achieved if the single δ-function in Eq. (1) is replaced by a symmetric pair, which corresponds to the equation introduced in Ref.[10], iu z + (1/2)ψ xx + [δ(x − L/2) + δ(x + L/2)] |ψ| 2 ψ = 0.(5)

We experimentally demonstrate that the orbital angular momentum (OAM) of a second harmonic (SH) beam, generated within twisted nonlinear photonic crystals, depends both on the OAM of the input pump beam and on the quasi-angular momentum of the crystal. In addition, when the pump's radial index is zero, the radial index of the SH beam is equal to that of the nonlinear crystal. Furthermore, by mixing two noncollinear pump beams in this crystal, we generate, in addition to the SH beams, a new "virtual beam" having multiple values of OAM that are determined by the nonlinear process.

We present experimentally the control of free space propagation of an Airy beam. This beam is generated by a nonlinear wave mixing process in an asymmetrically poled nonlinear photonic crystal. Changing the quasi-phase matching conditions, e.g., the crystal temperature or pump wavelength, alters the location of the Airy beam peak intensity along the same curved trajectory. We explain that the variation in the beam shape is caused by noncollinear interactions. Owing to the highly asymmetric shape of nonlinear crystal in the Fourier space, these noncollinear interactions are still relatively efficient for positive ͑nonzero͒ phase mismatch.