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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We present a general method for the design of 2-dimensional nonlinear photonic quasicrystals that can be utilized for the simultaneous phase-matching of arbitrary optical frequency-conversion processes. The proposed scheme-based on the generalized dual-grid method that is used for constructing tiling models of quasicrystals-gives complete design flexibility, removing any constraints imposed by previous approaches. As an example we demonstrate the design of a color fan-a nonlinear photonic quasicrystal whose input is a single wave at frequency ω and whose output consists of the second, third, and fourth harmonics of ω, each in a different spatial direction.PACS numbers: 42.65. Ky, 42.70.Mp, 61.44.Br, 42.79.Nv The problem of phase matching in the interaction of light waves in nonlinear dielectrics became immediately evident as the first theories describing such interaction were developed [1]. Put simply, nonlinear interaction is severely constrained in dispersive materials because the interacting photons must conserve their total energy and momentum. Even the slightest wave-vector mismatch appears as an oscillating phase that averages out the outgoing waves, hence the term "phase mismatch". One approach for treating the problem uses the birefringent properties of specific materials and by playing with the polarizations of the interacting waves achieves phase matching [2,3]. A second approach, suggested over 4 decades ago [1,4] and known today as "quasi-phasematching", is to modulate the sign of the relevant component(s) of the nonlinear dielectric tensor at the period of the oscillating mismatched phase thereby undoing the averaging. Quasi-phase-matching has been generalized from simple 1-dimensional periodic modulation [5] to 2-dimensional periodic modulation [6,7,8,9, 10] as well as 1-dimensional quasiperiodic modulation [11,12,13,14], allowing greater flexibility in phase-matching multiple frequency-conversion processes within the same photonic crystal. Here we present the full generalization of the method that enables the design of nonlinear photonic crystals that can simultaneously phase-match any arbitrary set of frequency-conversion processes in any spatial direction. This design flexibility is ideal for the realization of elaborate multi-step cascading effects [15,16], as demonstrated by the color fan example (Fig. 1) at the end of this article.To understand how the method works it is convenient to adopt the view taken in condensed matter systems. Recall that momentum conservation is a direct consequence of having continuous translation symmetry. In crystals, whether periodic or not, continuous translation symmetry is broken, and momentum conservation is re- placed by the less-restrictive conservation law of crystalmomentum. The total momentum of any set of interacting particles in a crystal-whether they are electrons, phonons, or photons-need only be conserved to within a wave vector from the reciprocal lattice of the crystal, giving rise to so-called umklapp processes. Thus, all one needs to do is to ...

We present a geometrical representation of sum frequency generation process in the undepleted pump approximation. The analogy of such dynamics with the known optical Bloch equations is discussed. We use this analogy to present a novel technique for the achievement of both high efficiency and large bandwidth in a sum frequency conversion processes using adiabatic inversion scheme, adapted from NMR and light-matter interaction. The adiabatic constraints are derived in this context. Last, this adiabatic frequency conversion scheme is realized experimentally by a proper design of adiabatic aperiodically poled KTP device, using quasi phased matched method. In the experiments we achieved high efficiency signal to idler conversion over a bandwidth of 140nm.PACS numbers: 42.65. Ky, 42.25.Fx, 42.70.Qs The generation of tunable frequency optical radiation typically relies on nonlinear frequency conversion in crystals. In this process, light of two frequencies is mixed in a nonlinear crystal, resulting in the generation of a third color with their sum or difference frequency. These three-wave mixing processes, also known as frequency up-conversion or frequency down-conversion are typically very sensitive to the incoming frequencies, owing to lack of phase matching of the propagating waves. Thus, angle, temperature or other tuning mechanisms are needed to support efficient frequency conversion. This difficulty is of particular importance when trying to efficiently convert broadband optical signals, since simultaneous phase matching of a broad frequency range is difficult.Solving the general form of the wave equations governing three wave mixing processes in nonlinear process is not an easy task. Under certain conditions these can be simplified to three nonlinear coupled equations. Further simplification can be applied when one incoming wave (termed pump) is much stronger than the other two. In the "undepleted pump" approximation, two linear coupled equation are obtained rather than three nonlinear ones [1]. In the case of sum frequency generation (SFG) process, this simplified system possesses SU(2) symmetry, sharing its dynamical behavior with other two states systems, such as nuclear magnetic resonance (NMR) or the interaction of coherent light with a two-level atom. In this letter we explore the dynamical symmetry of SFG process in analogy with the well known two level system dynamics [2]. We also apply a geometrical visualization using the approach presented by Bloch [3] and Feynman et al. [4] in NMR and light-matter interaction, respectively. The simple vector form of the coupling equation allows for new physical insight into the problem of fre- * Electronic address: Haim.suchowski@weizmann.ac.il quency conversion, enabling a more intuitive understanding of the effects of spatially varying coupling and phase mismatch. The utility of this approach is demonstrated by introducing a robust, highly efficient broadband color conversion scheme, based on an equivalent mechanism for achieving full population inversion in at...