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Phase mismatch and group velocity mismatch are two dispersion effects which limit the effectiveness of optical frequency conversion processes. Here we propose a condition for a high harmonic generation in which a phase mismatch and a group-velocity mismatch work together through constructive inter sub-cycle interference to provide an efficient, phase-matched optical frequency conversion. This condition, which we call an anomalous phase matching (APM) can be valuable to the emerging extreme non-linear optics regime of a long wavelength drive field and a high gas pressure. The APM condition is fulfilled once both phase mismatch and group velocity mismatch (GVM) are present in the optical system, which is the most general case to consider. The effect of a phase mismatch is that after a propagation of one coherence length the emission start to destruct coherently, leading to a decrease in the up-conversion efficiency. The effect of the GVM is that after a particular harmonic order is being generated by the fundamental pulse, it starts to drift away from its creation time in the reference frame moving with the pump pulse, due to different group velocities between the pump and that harmonic. If a harmonic emission arrives, due to GVM, to an emission with the opposite polarity after traversing a coherence length the interference of both emissions would be constructive. Actually, this condition ensures constructive interference over the whole interaction length allowing for an increased HHG flux. We derive a mathematical condition for an APM and show numerically that APM can be achieved and wavelength-tuned in several different setups: by varying the peak intensity of the pump laser in a pre-ionized gaseous medium, by varying the wave-guide radius in a hollow wave guide filled with a neutral gas and by varying the pressure of a neutral gas in a gas cell at moderate pressures and long fundamental beam wavelengths.

Symmetry-based analysis is very useful in many fields of science. In the case of time-dependent systems, the symmetries of interest are dynamical, and are maintained during the system's evolution in time. Interestingly, such symmetries may manifest over several length scales, e.g. both microscopically and macroscopically. For example, in high harmonic generation (HHG), micro-scale dynamical symmetries (DSs) yield selection rules that forbid some harmonic orders and determine the polarizations of the allowed harmonics [1], as was formulated recently in a consistent group theory [2]. Macroscopically, DSs of the wave equation may induce selection rules with respect to phase-matching conditions and conservation of angular or linear momentum [3]. However, the DSs of these different length scales are generally thought to be non-interacting, and have so far only been described separately. Here we formulate and explore plethora of multi-scale dynamical symmetries (MSDS), where the symmetry operations involve both microscopic and macroscopic length scales. We derive all possible MSDS operations and their resulting selection rules in HHG, and experimentally demonstrate non-trivial selection rules of the new theory.

We propose a simple quasi-phase-matching (QPM) scheme in high-order harmonic generation (HHG) based on the approximately linear dependence of the phase mismatch on harmonic order. With this scheme, essentially all the harmonic orders are phase matched simultaneously, with harmonic order $q$ experiencing a $q\mathrm{th}$-order QPM. We validate this proposal by simulations using a semiclassical numerical model of HHG.

We suggest a macroscopic scheme for generating circularly polarized high harmonics. The scheme uses a superposition of circularly polarized pump beams with different spatial modes, realizing a perturbed linearly polarized field which rotates along the propagation axis due to virtual circular birefringence. This method provides a spectral selectivity for the harmonic orders which are generated efficiently and for their spacing. The scheme is investigated numerically with a good agreement with the theory.

In this tutorial we discuss the macroscopic aspects of the extreme nonlinear frequency conversion process of high-order harmonic generation. For most, this entails describing the phase mismatch of the process and the means to mitigate it by tuning global parameters or by modulations perturbing the process. Additionally, we consider applications of phase matching in general for probing and for controlling different parameters of the emitted radiation.

High harmonic generation has been considered for many years to be a coherent source of upconverted laser radiation. We propose a method for the generation of essentially incoherent high harmonic generation with a controlled degree of temporal coherence, obeying random-walk statistics. This is achieved by a unique combination of single-atom and macroscopic effects involving preparation of the medium in an excited state with coordinate-dependent population inversion and using an intense off-resonant and phase-mismatched pumping.

High harmonic generation (HHG) is an extreme nonlinear frequency up-conversion process during which extremely short duration optical pulses at very short wavelengths are emitted. A major concern of HHG is the small conversion efficiency at the single emitter level. Thus ensuring that the emission at different locations are emitted in phase is crucial. At high pump intensities it is impossible to phase match the radiation without reverting to ordered modulations of either the medium or the pump field itself, a technique known as Quasi-Phase-Matching (QPM). To date, demonstrated QPM techniques of HHG were usually complicated and/or lacked tunability. Here we demonstrate experimentally a relatively simple, highly and easily tunable QPM technique by using a structured pump beam made of the interference of different spatial optical modes. With this technique we demonstrate on-the-fly, tunable quasi-phase-matching of harmonic orders 25 to 39 with up to 30 fold enhancement of the emission., 13 pages, 4 figures

We propose a paradigm for macroscopic control of high-order harmonic generation by modulating the bound-state population of the medium atoms. A unique result of this scheme is that apart from regular spatial quasi-phase-matching (QPM), also purely temporal QPM of the emitted radiation can be established. Our simulations demonstrate temporal QPM by inducing homogenous Rabi oscillations in the medium and also spatial QPM by creating a grating of population inversion using the process of rapid adiabatic passage. In the simulations a scaled version of high-order harmonic generation is used: a far off-resonance $2.6\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ source generates UV-visible high-order harmonics from alkali-metal-atom vapor, while a resonant near IR source is used to coherently control the medium.

It is theoretically shown that coherent control of atomic bound states can achieve macroscopic control over the process of High-order Harmonic Generation. In particular purely temporal Quasi-Phase-Matching is numerically demonstrated.

The two main mechanisms of a periodic density modulation relevant to nonlinear optical conversion in a gas medium are spatial modulations of the index of refraction and of the number of emitters. For a one-dimensional model neglecting focusing and using a constant ambient pressure, it is shown theoretically and demonstrated numerically that the effects of these two mechanisms during frequency conversion cancel each other exactly. Under the considered conditions, this makes density modulation inefficient for quasi-phase-matching an optical frequency conversion process. This result is particularly relevant for high-order harmonic generation.

We present a temporal extension of quasi-phase-matching of a nonlinear optical process. This allows conserving both momentum and energy over macroscopic length and time scales. This treatment is relevant for any dynamic quasi-phase-matching scheme.

We develop a description of nonlinear optics where quasi-phase-matching of both momentum and energy mismatch can be implemented by employing both spatial and temporal modulations. This generalized quasi-phase-matching technique could allow unique applications.

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-μm wavelength, in an extended, high-pressure, 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.

The generation of attosecond-duration light pulses using the high-order harmonic generation process is a rapidly evolving area of research. In this work, we combine experimental measurements with careful numerical analysis, to demonstrate that even relatively long-duration, 15 fs, carrier-envelope-phase (CEP) unstabilized near-infrared (NIR) pulses can generate isolated attosecond extreme-ultraviolet (EUV) pulses by the dynamically-changing phase matching conditions in a hollow-core waveguide geometry. The measurements are made using the laser-assisted photoelectric effect to cross-correlate the EUV pulse with the NIR pulse. A FROG CRAB analysis of the resulting traces (photoelectron signal versus photoelectron energy and EUV-NIR delay) is performed using a generalized projections (GP) algorithm, adapted for a wide-angle photoelectron detection geometry and non-CEP stabilized driving laser pulses. In addition, we performed direct FROG CRAB simulations under the same conditions. Such direct simulations allow more freedom to explore the effect of specific pulse parameters on FROG CRAB traces than is possible using the automated GP retrieval algorithm. Our analysis shows that an isolated pulse with duration of approximately 200 attoseconds can result from CEP unstabilized, high intensity approximately 15 fs multi-cycle driving pulses coupled into a hollow-core waveguide filled with low-pressure Argon gas. These are significantly longer driving pulses than used in other experimental implementations of isolated attosecond pulses.

The utility of quasiperiodic and random quasi-phase matching is demonstrated theoretically for shaping the spectral emission of high harmonic generation. Demonstrated are simultaneous enhancement of arbitrary spectral regions and enhancement of an extremely wide bandwidth.

We temporally characterize 200 attosecond FWHM pulses created by high harmonic generation of 15 femtosecond pulses. Simulations of the interferometric two-color crosscorrelation data and an iterative algorithm were used to extract the pulse duration.

It is shown theoretically that second harmonic generation can be quasi-phase-matched by using a pump beam consisting of a forward propagating field and a counterpropagating pulse train. The counterpropagating setup can also be used for direct measurement of the coherence length of the nonlinear process which can determine the dispersion properties of the medium.

All optical processes, e.g. deflection and polarization rotation, can be realized using cascaded χ(2): χ(2)interactions in one-dimensional and two-dimensional nonlinear photonic crystals. Collinear and non-collinear interactions in periodically- and quasi-periodically-modulated crystals are studied.

We recently described a general solution to the phase-matching problem that arises when one wishes to perform an arbitrary number of nonlinear optical processes in a single medium [Phys. Rev. Lett.95, 133901 (2005)]. Here we outline in detail the implementation of the solution for a one-dimensional photonic quasicrystal, which acts as a simultaneous frequency doubler for three independent optical beams. We confirm this solution experimentally using an electric-field poled KTiOPO4 crystal. In optimizing the device, we find—contrary to common practice—that simple duty cycles of 100% and 0% may yield the highest efficiencies, and we show that our device is more efficient than a comparable device based on periodic quasi-phase matching.

We show that a sawtooth phase-modulation is the optimal profile for grating assisted phase matching (GAPM). Perfect (sharp) sawtooth modulation fully corrects the phase-mismatch, exhibiting conversion equal to conventional phase matching, while smoothened, approximate sawtooth structures are more efficient than sinusoidal or square GAPM modulations that were previously studied. As an example, we demonstrate numerically optically-induced sawtooth GAPM for high harmonic generation. Sawtooth GAPM is the most efficient method for increasing the conversion efficiency of high harmonic generation through quasi-phase-matching, with an ultimate efficiency that closely matches the ideal phase-matching case.

Quasi-phase matching schemes employing quasi-periodic or random spatial modulations, previously applied to perturbative nonlinear optics, are demonstrated theoretically for the extreme nonlinear optical process of high harmonic generation. We show that quasi-periodic quasi-phase matching of high harmonic generation can be used for simultaneous enhancement of arbitrarily chosen spectral regions. We also demonstrate enhancement of a single extremely wide bandwidth using random quasi-phase matching.

We demonstrate that using longer-wavelength driving light in high-order harmonic generation requires moderately ionized larger-density atoms to perfectly phase match the process, scaling conventional phase matching at low ionization to extremely high photon energies.

A quasi-phase-matching method where the modulation has a sawtooth profile and the phase-mismatch is almost fully corrected is suggested. Schemes for implementing sawtooth grating-assisted-phase-matching in low-order as well as high-order harmonic generation are proposed.

We generate fully spatially coherent, ultrafast soft x-ray beams in the water window region of the spectrum using phase matched high harmonic upconversion of a 2 µm driving laser.