Theory Of High-order Harmonic Generation Research Articles

High-order harmonic generation by aligned homonuclear diatomic cations.

We introduce the theory of high-order harmonic generation by aligned homonuclear diatomic cations using a strong-field approximation. The target cation is represented as a system which consists of two atomic (ionic) centres and one active electron, while the driving field is either a monochromatic or bichromatic field. For a linearly polarised driving field, we investigate the differences between the harmonic spectra obtained with a neutral molecule and the corresponding molecular cation. Due to the larger ionisation potential, the molecular cations can withstand much higher laser-field intensity than the corresponding neutral molecule before the saturation effects become significant. This allows one to produce high-order harmonics with energy in the water-window interval or beyond. Also, the harmonic spectrum provides information about the structure of the highest-occupied molecular orbital. In order to obtain elliptically polarised harmonics, we suggest that an orthogonally polarised two-colour field is employed as a driving field. In this case, we analyse the harmonic ellipticity as a function of the relative orientation of the cation in the laser field. We show that the regions with large harmonic ellipticity in the harmonic energy-orientation angle plane are the broadest for cations whose molecular orbital does not have a nodal plane. Finally, we show that the molecular cations exposed to an orthogonally polarised two-colour field represent an excellent setup for the production of elliptically polarised attosecond pulses with a duration shorter than 100 as.

Macroscopic effects in high-order harmonic generation - a focal-averaging method based on the integral solution of the wave equation.

A macroscopic theory of high-order harmonic generation (HHG) is presented, which applies a focal-averaging method based on the integral solution of the wave equation. The macroscopic high-harmonic yield is the coherent superposition of the single-atom contributions of all atoms of the generating medium, which are positioned at different spatial points of the laser focus and exposed to the space-time-dependent laser pulse. The HHG spectrum obtained in our macroscopic simulations is qualitatively different from the one obtained using the microscopic or single-atom theory of HHG. Coherent intensity focal averaging, the simpler and more approximate of two methods we introduced, gives the spectrum which forms a declining plateau with the same cutoff position as that of the microscopic spectrum. The second, more precise method, which we call coherent spatio-temporal focal averaging, shows that it is possible, changing the macroscopic conditions, to obtain an observable peak in the harmonic spectrum at an energy much lower than the microscopic cutoff energy. Generally, the high-harmonic yield appears to be dominated by the contributions of laser-pulse spatio-temporal regions with lower intensities as well as by interference, so that the high-energy plateau and its sharp cutoff are quenched in the theoretical simulation and, presumably, in the experiment. The height and position of this peak strongly depend on the macroscopic conditions. We confirmed these findings by applying our macroscopic theory to simulate two recent experiments with mid-infrared laser fields, one with a linearly polarized field and the other one with a bicircular field.

Time and space waveform optimization to extend the harmonic cutoff and to produce the water window single attosecond pulse

Abstract Based on the three-step theory of high-order harmonic generation, the harmonic cutoff is very sensitive to the few-cycle laser waveform in both time and space regions. Therefore, in this paper, we propose the method to control the harmonic cutoff and to produce the water window attosecond pulse through the optimization of time and space waveform. It is found that, in the time region, by properly choosing the delay and phase of the few-cycle two-color pulse, not only the harmonic intensity is enhanced, but also the quantum path of the harmonic emission can be controlled. Further, with the introduction of the 3rd pulse (i.e., the infrared pulse or the unipolar pulse), the harmonic cutoff from the single harmonic emission peak can be extended, showing a water window harmonic plateau. In the space region, by using the positive spatial inhomogeneous effect, the harmonic cutoff from the basic two-color waveform can also be extended, which leads to a water window spectral continuum. Finally, by Fourier transformation of harmonics during the water window region, the ultrashort single 29 as pulses can be obtained.

Ultrahigh-order harmonic generation in the subnanometer wavelength range: the role of finite atomic size

As shown by recent theoretical studies, intense ultrafast laser sources of long-wave infrared range are capable, in principle, of generating ultrahigh-order harmonics in the wavelength range of the atomic or even subatomic scale. Under these conditions the atom as an elementary emitter can no longer be considered within the framework of the point dipole model generally accepted in the theory of high-order harmonic generation. It can be expected that the non-pointness of an atomic dipole will lead to a change in both the power of the radiation emitted by it and its directivity pattern. In this paper, we study these effects in detail using the example of the hydrogen atom. The analysis was carried out within the widely used recollision model, according to which the high-frequency dipole moment responsible for the generation of high harmonics by an atomic system is induced as a result of the interference of the de Broglie wave of the recolliding electron with the wave function of the atomic bound state. Taking into account the non-pointness of the induced dipole, the dependences of the power and directivity pattern of its radiation on the wavelength of the emitted photon are found. In particular, a universal atomic dipole non-pointness factor is obtained in an explicit form, which depends on the wavelength and emission angle of the harmonic photon and makes it possible to calculate the frequency and angular characteristics of the emitted harmonics based on the results obtained in the point dipole approximation.

Orientation dependences of high-order harmonic generation in H2 and H2+ molecules

We propose a modified version for the strong field theory of high-order harmonic generation (HHG) in molecules and we apply it to analyze the orientation-dependences of HHG from the hydrogen molecule and it molecular ion H2+ when exposed to linearly polarized laser light. Our model predicts the presence of clear interference minima in the spectra of the high-order harmonics. We further exploit the orientation-dependence of the interference minima’s position and the harmonics polarization properties through proposed model. We found that the position of the interference minima are associated with the polarization properties of harmonics.

Time-dependent density-functional theory of high-order harmonic generation from noble-gas atoms driven by orthogonally polarized two-color laser fields

An orthogonally polarized two-color (OTC) field of frequency ratio $1:2$ is known to generate both odd and even harmonic spectra in the mutually orthogonal directions of the fundamental and the second-harmonic fields, respectively. Recollision angles of an ionized electron driven back to its parent ion by an OTC field are inferred from the intensity ratio of odd and even harmonics, where the relative phase delays between the two-color fields can be used in time-resolved measurements on a subcycle time scale. In this work, we present the ab initio, many-electron calculation of time-resolved recollision angles of noble-gas atoms driven by an OTC field, based on the time-dependent density-functional theory. Our results agree well with experimental data by Shafir et al. in Nat. Phys. 5, 412 (2009) of helium atom and New J. Phys. 12, 073032 (2010) of neon atom.

Quantum theory of radiation by nonstationary systems with application to high-order harmonic generation

In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.

Quantum-orbit analysis of high-order harmonic generation by bicircular field

ABSTRACTQuantum-orbit theory of high-order harmonic generation (HHG) by bicircular laser field is presented. HHG is a strong laser-field-induced process in which the energy absorbed from the laser field is emitted in the form of a high-energy photon. This process can be described using the strong-field approximation and its approximation – the quantum-orbit theory. We develop a classification of quantum orbits for HHG by bicircular field which consists of two coplanar counter-rotating circularly polarized field of frequencies and , where r and s are integers and ω is a fundamental frequency. Analysis of the contributions of particular quantum orbits to high-harmonic intensity enables a better understanding of the HHG process. The cases of the ω– and ω– bicircular fields and of the atoms having the s and the p ground state are analysed in detail. Particular attention is devoted to the influence of the ratio of the intensities and of the bicircular field components. For inert gases having the p ground state the asymmetry in the emission of the left- and right-circularly polarized harmonics can be large. This is explained comparing the partial harmonic intensities for HHG from the ground state having the magnetic quantum number m=+1 and m=−1 and analysing the contributions of particular quantum orbits and the corresponding electron trajectories and velocities. The contribution of the shortest quantum orbit is dominant. It was found that the electron velocity at the ionization time, which still allows the return of the electron to the parent ion, determines the height of the high-energy HHG plateau. For this velocity is large which, in the case of the p ground state, leads to a large helicity asymmetry parameter. On the other hand, for this velocity is small and the intensity of the high-energy plateau harmonics is high.

Theory of high-order harmonic generation for gapless graphene

We study the high-harmonic spectrum emitted by a single-layer graphene, irradiated by an ultrashort intense infrared laser pulse. We show the emergence of the typical non-perturbative spectral features, harmonic plateau and cut-off, for mid-infrared driving fields, at fluences below the damage threshold. In contrast to previous works, using THz drivings, we demonstrate that the harmonic cut-off frequency saturates with the intensity. Our results are derived from the numerical integration of the time-dependent Schrödinger equation using a nearest neighbor tight-binding description of graphene. We also develop a saddle-point analysis that reveals a mechanism for harmonic emission in graphene different from that reported in atoms, molecules and finite gap solids. In graphene, the first step is initiated by the non-diabatic crossing of the valence band electron trajectories through the Dirac points, instead of tunneling ionization/excitation. We include a complete identification of the trajectories contributing to any particular high harmonic and reproduce the harmonic cut-off scaling with the driving intensity.

High-order harmonic generation by polyatomic molecules

We present a theory of high-order harmonic generation by arbitrary polyatomic molecules based on the molecular strong-field approximation (MSFA) in the framework of the S-matrix theory. A polyatomic molecule is modeled by an (N + 1)-particle system, which consists of N heavy atomic (ionic) centers and an electron. We derived various versions (with or without the dressing of the initial and/or final molecular state) of the MSFA. The general expression for the T-matrix element takes a simple form for neutral polyatomic molecules. We show the existence of the interference minima in the harmonic spectrum and explain these minima as a multiple-slit type of interference. This is illustrated by numerical examples for the nitrous oxide (N2O) molecule exposed to strong linearly polarized laser field.

Strong-field-approximation theory of high-order harmonic generation by polyatomic molecules

A theory of high-order harmonic generation by arbitrary polyatomic molecules is introduced. A polyatomic molecule is modeled by an $(N+1)$-particle system, which consists of $N$ heavy atomic (ionic) centers and an electron. After the separation of the center-of-mass coordinate, the dynamics of this system is reduced to the relative electronic and nuclear coordinates. Various versions (with or without the dressing of the initial and/or final molecular state) of the molecular strong-field approximation are introduced. For neutral polyatomic molecules the derived expression for the $T$-matrix element takes a simple form. The interference minima in the harmonic spectrum are explained as a multiple-slit type of interference. This is illustrated by numerical examples for the ozone (${\mathrm{O}}_{3}$) and carbon dioxide (${\mathrm{CO}}_{2}$) molecules.

Self-consistent theory of high-order harmonic generation by relativistic plasma mirror

We present a self-consistent semianalytical model of the relativistic plasma mirror, based on the exact computation of the laser-driven electron surface oscillations within the cold-fluid approximation. Valid for arbitrary solid densities, laser incidence angle, and a large set of laser intensities (10(18)-10(21) W/cm(2)), the model unravels different regimes of harmonic generation. In particular, it is found that efficient conversion of p-polarized laser pulses into high-order harmonics well above the plasma frequency requires either high laser intensities, low plasma densities, or incidence angles larger than a threshold value. This critical angle corresponds to a transition between a regime where the electron surface dynamics is mostly governed by the laser J×B force and a "cyclotron Brunel" regime, where electrons perform many cyclotron gyrations when moving into the vacuum. Under conditions relevant to current laser experiments, the latter regime gives rise to nonmonotonic variations of the harmonic yield with the laser field. Our predictions are supported by an extensive parametric study performed with highly resolved one-dimensional particle-in-cell simulations.

Circularly polarized high harmonics generated by a bicircular field from inert atomic gases in thepstate: A tool for exploring chirality-sensitive processes

$S$-matrix theory of high-order harmonic generation (HHG) is generalized to multielectron atoms. In the multielectron case the harmonic power is expressed via a coherent sum of the time-dependent dipoles, while for the one-electron models a corresponding incoherent sum appears. This difference is important for the inert atomic gases having a $p$ ground state as used in a recent HHG experiment with a bicircular field [Nat. Photonics 9, 99 (2015)]. We investigate HHG by such a bicircular field, which consists of two coplanar counter-rotating circularly polarized fields of frequency $r\ensuremath{\omega}$ and $s\ensuremath{\omega}$. Selection rules for HHG by a bicircular field are analyzed from the aspects of dynamical symmetry of the system, conservation of the projection of the angular momentum on a fixed quantization axis, and the quantum number of the initial and final atomic ground states. A distinction is made between the selection rules for atoms with closed [J. Phys. B 48, 171001 (2015)] and nonclosed shells. An asymmetry in emission of the left- and right-circularly polarized harmonics is found and explained by using a semiclassical model and the electron probability currents which are related to a nonzero magnetic quantum number. This asymmetry can be important for the application of such harmonics to the exploration of chirality-sensitive processes and for generation of elliptic or even circular attosecond pulse trains. Such attosecond pulse trains are analyzed for longer wavelengths than in Opt. Lett. 40, 2381 (2015), and for various field-component intensities.

High-order harmonic generation by a bichromatic elliptically polarized field: conservation of angular momentum

We present a theory of high-order harmonic generation by a bichromatic elliptically polarized laser field which consists of two coplanar components having the frequencies and (r and s are integers) and is defined in the xy plane. Laser and harmonic fields are decomposed in the components having opposite helicities. Using the conservation laws for energy and projection of the total angular momentum of atom and laser and harmonic photons on the z axis we have derived a general selection rule. This rule reduces to the known result in the case of bicircular field which consists of two counter-rotating circularly polarized fields. We apply our results to explain recent experiment by Fleischer et al 2014 (Nature Photonics 8 543).

Quantitative rescattering theory of high-order harmonic generation for polyatomic molecules

We report applications of the quantitative rescattering theory (QRS) for calculation of high-order-harmonic generation (HHG) from polyatomic molecules in ultrashort linearly polarized intense laser pulses, using the example of the CCl4 molecule. In particular, we present a detailed analysis and a treatment for the phase of the electron returning wave packet, which recollides with the parent molecular ion to emit high-energy photons. Our results show that Cooper-type minimum structures in the molecular photoionization cross section lead to quite distinguishable minima in the HHG spectra, even for unaligned polyatomic molecules.

Dressed-bound-state molecular strong-field approximation: application to high-order harmonic generation by heteronuclear diatomic molecules

A theory of high-order harmonic generation by heteronuclear diatomic molecules exposed to an elliptically polarized laser field is presented. This theory is based on the molecular strong-field approximation, which takes into account the dressing of the initial and final molecular bound states caused by the laser field. The laser-field-caused Stark shift of the effective ionization potential of the highest occupied molecular orbital changes the structure of the harmonic spectra. Both the even and odd harmonics appear in the harmonic spectra due to the broken inversion symmetry of the heteronuclear diatomic molecule. An interference minima condition that is valid for arbitrary heteronuclear diatomic molecules is derived. Our results are illustrated using examples of CO and NO molecules.

Adiabatic theory of high-order harmonic generation: One-dimensional zero-range-potential model

The adiabatic theory of ionization of atoms by intense few-cycle laser pulses, whose development was initiated recently [Tolstikhin et al., Phys. Rev. A 81, 033415 (2010)], is applied to the study of high-order harmonic generation (HHG). The same one-dimensional zero-range-potential model is considered. The asymptotic formula for the HHG spectrum in this model is obtained in the leading-order adiabatic approximation. This asymptotics is uniform in terms of the photon energy in the cutoff region. It is shown that the factorization formula for the HHG spectrum, first proposed by Morishita et al. [Phys. Rev. Lett. 100, 013903 (2008)] on the basis of numerical solution of the time-dependent Schr\odinger equation, results from the adiabatic theory, and the structure of the rescattering photoelectron wave-packet factor is clarified. The theory is illustrated by calculations.

A scaling law of high-order harmonic generation

Using a nonperturbative quantum electrodynamics theory of high-order harmonic generation (HHG), a scaling law of HHG is established. The scaling law states that when the atomic binding energy Eb, the wavelength λ and the intensity I of the laser field change simultaneously to kEb, λ/k, and k3I, respectively. The characteristics of the HHG spectrum remain unchanged, while the harmonic yield is enhanced k3 times. That HHG obeys the same scaling law with above-threshold ionization is a solid proof of the fact that the two physical processes have similar physical mechanisms. The variation of integrated harmonic yields is also discussed.

Pressure-induced phase matching in high-order harmonic generation

We present an alternative explanation of the high-order-harmonic-generation experimental results published recently by Seres et al. [Nature Phys. 6, 455 (2010)]. We show that the physical interpretation can be comprehensively done in the frame of classical theory of high-order harmonic generation without referring to a parametric effect in the XUV domain. The experimental conditions explored by Seres et al. indeed correspond to the case of long-pulse, low-infrared-energy laser beams for which tight focusing is necessary to reach the minimum intensity required for high atomic response. The positive atomic dispersion can compensate for the Gouy phase and explains the behavior of the experimental variation of the harmonic signal presented. We will show that our interpretation explains not only the global behavior of the curves but also the second-order variation of the signal as a function of experimental parameters.

Ellipticity and the offset angle of high harmonics generated by homonuclear diatomic molecules

In our recent paper (2010 Phys. Rev. A 82 023412) we introduced a theory of high-order harmonic generation by diatomic molecules exposed to an elliptically polarized laser field and have shown that the nth harmonic emission rate has contributions of the components of the T-matrix element in the direction of the laser-field polarization and in the direction perpendicular to it. Using both components of the T-matrix element we now develop a theoretical approach for calculating ellipticity and the offset angle of high harmonics. We show that the emitted harmonics generated by aligned molecules are elliptically polarized even if the applied field is linearly polarized. Using examples of N2, O2 and Ar2 molecules we show the existence of extrema and sudden changes of the harmonic ellipticity and the offset angle for particular molecular alignment and explain them by the destructive two-centre interference. Taking into account that the aligned molecules are an anisotropic medium for high harmonic generation, we introduce elliptic dichroism as a measure of this anisotropy, for both components of the T-matrix element. We propose that the measurement of the elliptic dichroism may reveal further information about the molecular structure.