We show how bright, tabletop, fully coherent hard X-ray beams can be generated through nonlinear upconversion of femtosecond laser light. By driving the high-order harmonic generation process using longer-wavelength midinfrared light, we show that, in theory, fully phase-matched frequency upconversion can extend into the hard X-ray region of the spectrum. We verify our scaling predictions experimentally by demonstrating phase matching in the soft X-ray region of the spectrum around 330 eV, using ultrafast driving laser pulses at 1.3-m wavelength, in an extended, highpressure, weakly ionized gas medium. We also show through calculations that scaling of the overall conversion efficiency is surprisingly favorable as the wavelength of the driving laser is increased, making tabletop, fully coherent, multi-keV X-ray sources feasible. The rapidly decreasing microscopic single-atom yield, predicted for harmonics driven by longer-wavelength lasers, is compensated macroscopically by an increased optimal pressure for phase matching and a rapidly decreasing reabsorption of the generated X-rays.coherent ͉ ultrafast ͉ extreme nonlinear optics A dvances in X-ray science and technology have resulted in breakthrough discoveries ranging from unraveling the structure of DNA and proteins to visualizing atoms, molecules, and materials at the nanoscale level. These continuing successes have spurred the development of X-ray free-electron laser sources that promise to create superexcited states of matter or to capture the structure of a single biomolecule. Another very exciting advance in X-ray science has been the ability to generate ultrafast (0.1-10 fs), coherent X-rays from a tabletop-scale apparatus, by using the extreme nonlinear optical process of high-order harmonic generation (HHG). In HHG, the output of a tabletop femtosecond laser is upconverted into the extreme-UV and soft X-ray regions of the spectrum. The unique characteristics of HHG soft X-rays have opened up many new scientific opportunities. The femtosecond-to-attosecond pulse duration has made it possible to capture the coupled motions of electrons, atoms, and molecules in real time (1-7). Moreover, the low divergence and capability to produce light with full spatial coherence (8) have enabled static and dynamic diffraction and imaging with resolutions of tens of nanometers (9, 10).The microscopic physics of the high harmonic generation process can be understood in terms of an intuitive recollision model (11,12). In this model, an electron is first ripped from an atom by the strong electric field of a focused laser beam. Once liberated, the electron is accelerated to high energies by the oscillating laser field and can violently recollide with its parent ion. Finally, if the electron recombines, any excess kinetic energy acquired in the external field is emitted as a high-energy photon. This simple physics can be used to determine a maximum photon energy that can be generated, or a so-called single-atom cutoff (12, 13):[1]Here, U p is the average kinetic energy of a ...

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 ...

Superoscillations are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies originated in quantum theory, but there were anticipations in radar and optics. The mathematical understanding—still being explored—recognises that functions are extremely small where they superoscillate; this has implications for information theory. Applications to optical vortices, sub-wavelength microscopy and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. This Roadmap surveys all these areas, providing background, current research, and anticipating future developments.

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