Theory of high-harmonic generation by low-frequency laser fields

Near-threshold high-order harmonic spectroscopy with aligned molecules

We study high-order harmonic generation in aligned molecules close to the ionization threshold. Two distinct contributions to the harmonic signal are observed, which show very different responses to molecular alignment and ellipticity of the driving field. We perform a classical electron trajectory analysis, taking into account the significant influence of the Coulomb potential on the strong-field-driven electron dynamics. The two contributions are related to primary ionization and excitation processes, offering a deeper understanding of the origin of high harmonics near the ionization threshold. This Letter shows that high-harmonic spectroscopy can be extended to the near-threshold spectral range, which is in general spectroscopically rich.

High-harmonic homodyne detection of the ultrafast dissociation of Br2 molecules

We report the time-resolved observation of the photodissociation of Br2 using high-harmonic generation (HHG) as a probe. The simultaneous measurement of the high-harmonic and ion yields shows that high harmonics generated by the electronically excited state interfere with harmonics generated by the ground state. The resulting homodyne effect provides a high sensitivity to the excited state dynamics. We present a simple theoretical model that accounts for the main observations. Our experiment paves the way towards the dynamic imaging of molecules using HHG.

Multichannel molecular high-order harmonic generation from asymmetric diatomic molecules

Multichannel molecular high-order harmonic generation (MHOHG) from a single electron asymmetric molecular ion HeH2+ is investigated numerically. It is found that considerable resonant excitation occurs by laser induced electron transfer (LIET) to neighboring ions and multiple frequency (fractional-order) harmonics are observed from the excited states shifted by some energy Δ from the main Nω energy harmonics. A time series analysis is used to confirm this MHOHG channel which is created by initial ionization from the excited state prepared by LIET and recombination to the neighboring ion at specific field phases, resulting in interference between recombination pathways from ground and excited states.

Interplay between group-delay-dispersion-induced polarization gating and ionization to generate isolated attosecond pulses from multicycle lasers

We implemented a new experimental scheme for the generation of single-shot extreme-UV continua that exploits a combination of transform-limited 15 fs, 800 nm pulses and chirped 35 fs, 800 nm pulses with orthogonal polarizations. Continua are interpreted as the formation of a single attosecond pulse and attributed to the interplay between polarization, ionization gating, and trajectory selection operated by suitable phase-matching conditions.

Extreme ultraviolet frequency comb metrology

The remarkable precision of frequency-comb (FC) lasers is transferred to the extreme ultraviolet (XUV, wavelengths shorter than 100 nm), a frequency region previously not accessible to these devices. A frequency comb at XUV wavelengths near 51 nm is generated by amplification and coherent up-conversion of a pair of pulses originating from a near-infrared femtosecond FC laser. The phase coherence of the source in the XUV is demonstrated using helium atoms as a ruler and phase detector. Signals in the form of stable Ramsey-like fringes with high contrast are observed when the FC laser is scanned over P states of helium, from which the absolute transition frequency in the XUV can be extracted. This procedure yields a (4)He ionization energy at h×5 945 204 212(6)  MHz, improved by nearly an order of magnitude in accuracy, thus challenging QED calculations of this two-electron system.

Strongly dispersive transient Bragg grating for high harmonics

We create a transient Bragg grating in a high-harmonic generation medium using two counterpropagating pulses. The Bragg grating disperses the harmonics in angle and can diffract a large bandwidth with temporal resolution limited only by the source size.

Controlling the interference of multiple molecular orbitals in high-harmonic generation

We demonstrate a new method to investigate the origin of spectral structures in high-harmonic generation. We report detailed measurements of high-harmonic spectra in aligned nitrogen and carbon dioxide molecules. Varying the wavelength and intensity of the generating laser field, we show that the minimum in aligned N2 molecules is nearly unaffected, whereas the minimum in aligned CO2 molecules shifts over more than 15 eV. Our quantitative analysis shows that both the interference of multiple orbitals and their structural characteristics affect the position of the minimum. Our method provides a simple approach to the investigation of the high-harmonic generation process in more complex molecules.

Mid-infrared modulated polarization gating for ultra-broadband supercontinuum generation

We propose a method to control the harmonic process by a mid-IR modulated polarization gating for the effective generation of an ultra-broadband supercontinuum in the neutral rare-gas media. Using a mid-IR polarization gating pulse modulated by a weaker 800-nm linearly polarized pulse, the ionization, acceleration and recombination steps in the harmonic process are simultaneously controlled, leading to the efficient generation of an ultra-broadband supercontinuum covered by the spectral range from ultraviolet to water window x ray under the low ionization rate. The right phase-matching technique is employed to macroscopically select the short quantum path of the supercontinuum, then isolated sub-100-as pulses with tunable wavelengths are directly obtained. This supercontinuum also supports the pulse duration far below one atomic unit of time.

High harmonic emission from a superposition of multiple unrelated frequency fields

We report observations and analysis of high harmonic generation driven by a superposition of fields at 1290 nm and 780 nm. These fields are not commensurate in frequency and the superposition leads to an increase in the yield of the mid-plateau harmonics of more than two orders of magnitude compared to using the 1290 nm field alone. Significant extension of the cut-off photon energy is seen even by adding only a small amount of the 780 nm field. These observations are explained by calculations performed in the strong field approximation. Most importantly we find that enhancement is found to arise as a consequence of both increased ionization in the sum-field and modification of the electron trajectories leading to an earlier return time. The enhanced yield even when using modest intensity fields of 5 x 10(13) Wcm(-2) is extended to the 80 eV range and is a promising route to provide a greater photon number for applications in XUV imaging and time-resolved experiments at a high repetition rate.

All-regions tunable high harmonic enhancement by a periodic static electric field

Simulations show that a static electric field periodically distributed in space can be used to control the production of coherent light by high-order harmonic generation in a wide spectral range covering extreme-ultraviolet and soft x-ray radiation. The radiation yield is selectively enhanced due to symmetry breaking induced by a static electric field on the interaction between the driving laser and the medium. The spectral position of the enhancement is tuned by varying the periodicity of the static electric field which matches twice the coherence length of the harmonics in the desired region. We find that the static electric field strength inducing enhancement decreases for shorter wavelengths and predict an increase of more than two orders of magnitude for harmonics in the water window spectral range with a static electric field as weak as 1.12 MV/cm.

Extension of high harmonic spectroscopy in molecules by a 1300 nm laser field

The emerging techniques of molecular spectroscopy by high order harmonic generation have hitherto been conducted only with Ti:Sapphire lasers which are restricted to molecules with high ionization potentials. In order to gain information on the molecular structure, a broad enough range of harmonics is required. This implies using high laser intensities which would saturate the ionization of most molecular systems of interest, e.g. organic molecules. Using a laser at 1300 nm, we are able to extend the technique to molecules with relatively low ionization potentials (approximately 11 eV), observing wide harmonic spectra reaching up to 60 eV. This energy range improves spatial resolution of the high harmonic spectroscopy to the point where interference minima in harmonic spectra of N(2)O and C(2)H(2) can be observed.

Spatial fingerprint of quantum path interferences in high order harmonic generation

We have spatially and spectrally resolved the high order harmonic emission from an argon gas target. Under proper phase matching conditions we were able to observe for the first time the spatial fine structure originating from the interference of the two shortest quantum paths in the harmonic beam. The structure can be explained by the intensity-dependent harmonic phase of the contributions from the two paths. The spatially and spectrally resolved measurements are consistent with previous spatially integrated results. Our measurement method represents a new tool to clearly distinguish between different interference effects and to potentially observe higher order trajectories in the future with improved detection sensitivity. Here, we demonstrate additional experimental evidence that the observed interference pattern is only due to quantum-path interferences and cannot be explained by a phase modulation effect. Our experimental results are fully supported by simulations using the strong field approximation and including propagation.

Role of ionization in orientation dependence of molecular high-order harmonic generation

We investigate the orientation dependence of high-order harmonic generation (HHG) from O(2) and CO(2) molecules using the strong-field approximation (SFA). Our simulations reveal the important modulation of the ionization to the HHG orientation dependence, especially at larger orientation angles. By virtue of a simplified model arising from the SFA, we show that this modulation can be read from the harmonic order where the HHG spectra at different orientation angles intersect. These results give suggestions on probing the molecular structure and dynamics using HHG.

Time-delay compensated monochromator for the spectral selection of extreme-ultraviolet high-order laser harmonics

The design and the characterization of a monochromator for the spectral selection of ultrashort high-order laser harmonics in the extreme ultraviolet are presented. The instrument adopts the double-grating configuration to preserve the length of the optical paths of different diffracted rays, without altering the extremely short duration of the pulse. The gratings are used in the off-plane mount to have high efficiency. The performances of the monochromator have been characterized in terms of spectral response, efficiency, photon flux, imaging properties, and temporal response. In particular, the temporal characterization of the harmonic pulses has been obtained using a cross-correlation method: Pulses as short as 8 fs have been measured at the output of the monochromators, confirming the effectiveness of the time-delay compensated configuration.

Generating single attosecond pulse using multi-cycle lasers in a polarization gate

We analyze the macroscopic effects which are responsible for producing clean isolated pulses lasting few hundreds of attoseconds when starting from multi-cycle fundamental pulses. In particular, we consider a polarization gating scheme and show that, at high fundamental peak intensities, in the range 0.7-1 PWcm(-2), it usually produces three-four main attosecond pulses of radiation at single dipole level, just located in the leading edge of the laser pulse. We describe the physical mechanisms contributing to the formation of a single attosecond pulse by using a three dimensional non-adiabatic model and a quantum trajectory phase calculation. An analysis of the scheme optimization and stability against various parameters is performed in view of an experimental scheme implementation.

Time-Resolved ESCA: a Novel Probe for Chemical Dynamics

Electron spectroscopy for chemical analysis (ESCA) is a well established tool for quantitative studies of the composition and the chemical environment of molecular systems. Recent developments in the generation and utilization of ultrashort X-ray pulses now add the dimension of time to this technique and will expand the possibilities of femtochemistry in terms of chemical selectivity, quality of information, and temporal resolution. The properties and capabilities of various X-ray pulse sources are discussed, along with their prospects for dynamical studies. Examples of time-resolved electron spectroscopy are presented in the femtosecond (1 fs = 10−15 s) as well as the attosecond (1 as = 10−18 s) regime, the latter marking the current ultimate limit for the time-resolution in pump-probe experiments.

Propagation of spatial coherence in fast pulses

Diffraction and interferometry with fast pulses are analyzed for the case that the fields are partially correlated in time and in space. This generalizes a previous work [Schoonover, J. Mod. Opt.55, 1541(2008)], where only the temporal correlations of pulsed fields were considered in a Young's interferometer. The meaning of the interferograms is addressed for measurements taken in the near, Fresnel, and far zones of the source. It is shown that single-shot measurements cannot generally be used to infer statistical properties of the source, rather, data averaged over many pulses must be used.

Efficient selection of single high-order harmonic caused by atomic autoionizing state influence

It is shown that the influence of atomic autoionizing states on the phase matching of high harmonic generation results in efficient selection of the single harmonic generated in the plateau region. The change of relative concentration of the atomic medium components and variation of the fundamental laser frequency can be used for tuning of the selected harmonic frequency. The possibility of achievement of contrast for the selected single harmonic of more than 10(4) at intensity conversion efficiency of approximately 10(-3) has been shown.

Wavelength scaling of high harmonic generation efficiency

Using longer wavelength laser drivers for high harmonic generation is desirable because the highest extreme ultraviolet frequency scales as the square of the wavelength. Recent numerical studies predict that high harmonic efficiency falls dramatically with increasing wavelength, with a very unfavorable lambda(-(5-6)) scaling. We performed an experimental study of the high harmonic yield over a wavelength range of 800-1850 nm. A thin gas jet was employed to minimize phase matching effects, and the laser intensity and focal spot size were kept constant as the wavelength was changed. Ion yield was simultaneously measured so that the total number of emitting atoms was known. We found that the scaling at constant laser intensity is lambda(-6.3+/-1.1) in Xe and lambda(-6.5+/-1.1) in Kr over the wavelength range of 800-1850 nm, somewhat worse than the theoretical predictions.

Dynamic chirp control and pulse compression for attosecond high-order harmonic emission

We propose a scheme to compensate dynamically the intrinsic chirp of the attosecond harmonic pulses. By adding a weak second harmonic laser field to the driving laser field, the chirp compensation can be varied from the negative to the positive continuously by simply adjusting the relative time delay between the two-color pulses. Using this technique, the compensation of the negative chirp in harmonic emission is demonstrated experimentally for the first time and the nearly transform-limited attosecond pulse trains are obtained.

Intensity spikes in laser filamentation: diagnostics and application

We present calculations with subcycle precision of laser-driven filamentation in argon which show that the laser peak intensity can exceed the clamping intensity by a factor of 3 during recurring spikes which can last over several centimeters of propagation. The high intensity occurs during a few-femtosecond subpulse in the trailing edge of the main pulse and gives rise to isolated 500 attosecond, 2 pJ pulses which can be extracted from the filament. We also show that the high harmonic radiation emerging from the filament is an excellent diagnostic of the intensity spikes.

High harmonic interferometry of multi-electron dynamics in molecules

High harmonic emission occurs when an electron, liberated from a molecule by an incident intense laser field, gains energy from the field and recombines with the parent molecular ion. The emission provides a snapshot of the structure and dynamics of the recombining system, encoded in the amplitudes, phases and polarization of the harmonic light. Here we show with CO(2) molecules that high harmonic interferometry can retrieve this structural and dynamic information: by measuring the phases and amplitudes of the harmonic emission, we reveal 'fingerprints' of multiple molecular orbitals participating in the process and decode the underlying attosecond multi-electron dynamics, including the dynamics of electron rearrangement upon ionization. These findings establish high harmonic interferometry as an effective approach to resolving multi-electron dynamics with sub-Angström spatial resolution arising from the de Broglie wavelength of the recombining electron, and attosecond temporal resolution arising from the timescale of the recombination event.

Broadband water window supercontinuum generation with a tailored mid-IR pulse in neutral media

We propose a new (to our knowledge) scheme to generate a broadband water window supercontinuum in a neutral rare-gas media. Using a phase-stabilized few-cycle mid-IR pulse tailored by a much weaker 800 nm pulse, a supercontinuum from 265 to 425 eV is observed, and the ionization probability is below 0.1%. The nonionized media enable us to adopt the right phase-matching technique to select the short path of the supercontinuum, and an isolated 80 as pulse with a central wavelength of 3.2 nm is straightforwardly obtained.

Nuclear dynamics in polyatomic molecules and high-order harmonic generation

High-order harmonic generation in molecular gases is accompanied by short-time evolution of the nuclear vibrational wave function. Using normal coordinate representation, I derive a simple analytical theory of short-time autocorrelation functions and apply it to a test set of 15 small molecules. The results explain large isotope effects observed in CH4. At the harmonic cutoff in 800 nm driving field, nuclear dynamics reduces the emission intensity from NO and NO2 molecules by more than 50%. Autocorrelation functions are sensitive to the initial vibrational state, with the nodal structure of the initial vibrational wave packet reflected in the frequency spectrum of the harmonics.

Analytic scaling analysis of high harmonic generation conversion efficiency

Closed form expressions for the high harmonic generation (HHG) conversion efficiency are obtained for the plateau and cutoff regions. The presented formulas eliminate most of the computational complexity related to HHG simulations, and enable a detailed scaling analysis of HHG efficiency as a function of drive laser parameters and material properties. Moreover, in the total absence of any fitting procedure, the results show excellent agreement with experimental data reported in the literature. Thus, this paper opens new pathways for the global optimization problem of extreme ultraviolet (EUV) sources based on HHG.

Analytic description of the high-energy plateau in harmonic generation by atoms: can the harmonic power increase with increasing laser wavelengths?

A closed-form analytic formula for high-order harmonic generation (HHG) rates for atoms (that generalizes an HHG formula for negative ions [M. V. Frolov, J. Phys. B 42, 035601 (2009)10.1088/0953-4075/42/3/035601]) is used to study laser wavelength scaling of the HHG yield for harmonic energies in the cutoff region of the HHG plateau. We predict increases of the harmonic power for HHG by Ar, Kr, and Xe with increasing wavelength lambda over atom-specific intervals of lambda in the infrared region, lambda approximately (0.8-2.0) microm.

Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum

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-microm 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.

Harmonic generation beyond the Strong-Field Approximation: the physics behind the short-wave-infrared scaling laws

The physics of laser-mater interactions beyond the perturbative limit configures the field of extreme non-linear optics. Although most experiments have been done in the near infrared ( lambda <or= 1 microm), the situation is changing nowadays with the development of sources at longer wavelengths (<5 microm), opening new perspectives in the synthesis of shorter XUV attosecond pulses and higher frequencies. The theory of intense-field interactions is based either on the exact numerical integration of the time-dependent Schrödinger equation or in the development of models, mostly based on the strong-field approximation. Recent studies in the short-wave infrared show a divergence between the predictions of these models and the exact results. In this paper we will show that this discrepancy reveals the incompleteness of our present understanding of high-order harmonic generation. We discuss the physical grounds, provide a theoretical framework beyond the standard approximations and develop a compact approach that accounts for the correct scaling of the harmonic yield.

Multi-quantum-path interference in high harmonic generation driven by a chirped laser pulse

We theoretically investigate the quantum-path interference during the high harmonic generation in argon and observe the multi-quantum- path-interference (MQPI) effect related to the chirp of driving laser pulse. We successfully identify the interference originated from four significantly contributing quantum paths in the interaction of "isolated" atom with the driving laser field. Moreover, the MQPI effect is further demonstrated beyond the single atom response by spatial filtering which is used for angle-selective detection of transmitted light when three-dimensional propagation is considered. It implies that the role of high-order electron returning trajectories during high harmonic generation can be observed in the experiment.

Highly stable ultrabroadband mid-IR optical parametric chirped-pulse amplifier optimized for superfluorescence suppression

We present a 9 GW peak power, three-cycle, 2.2 microm optical parametric chirped-pulse amplification source with 1.5% rms energy and 150 mrad carrier envelope phase fluctuations. These characteristics, in addition to excellent beam, wavefront, and pulse quality, make the source suitable for long-wavelength-driven high-harmonic generation. High stability is achieved by careful optimization of superfluorescence suppression, enabling energy scaling.

Interferometry of attosecond pulse trains in the extreme ultraviolet wavelength region

The temporal coherence of an attosecond optical field in the extreme ultraviolet wavelength region can be defined in terms of the extent of interference in time domain. We successfully measured this phenomenon both with and without spectral decomposition. We also report the results of using this approach to directly observe both symmetry and symmetry breaking of interference fringes in an attosecond pulse train.

Probing molecular frame photoionization via laser generated high-order harmonics from aligned molecules

Present experiments cannot measure molecular frame photoelectron angular distributions (MFPAD) for ionization from the outermost valence orbitals of molecules. We show that the details of MFPAD can be retrieved with high-order harmonics generated by infrared lasers from aligned molecules. Using accurately calculated photoionization transition dipole moments for fixed-in-space molecules, we show that the dependence of the magnitude and phase of the high-order harmonics on the alignment angle of the molecules observed in recent experiments can be quantitatively reproduced. This result provides the needed theoretical basis for ultrafast dynamic chemical imaging using infrared laser pulses.

Characterizing isolated attosecond pulses from hollow-core waveguides using multi-cycle driving pulses

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.

Attosecond synchronization of high-order harmonics from midinfrared drivers

The group delay dispersion, also known as the attochirp, of high-order harmonics generated in gases has been identified as the main intrinsic limitation to the duration of Fourier-synthesized attosecond pulses. Theory implies that the attochirp, which is inversely proportional to the laser wavelength, can be decreased at longer wavelength. Here we report the first measurement of the wavelength dependence of the attochirp using an all-optical, in situ method [N. Dudovich, Nature Phys. 2, 781 (2006)10.1038/nphys434]. We show that a 2 microm driving wavelength reduces the attochirp with respect to 0.8 microm at comparable intensities.

Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period

We present the perfect waveform which, during a strong field interaction, generates the maximum possible electron recollision energy for any given oscillation period, over 3 times as high as that for a pure sinusoidal wave. This ideal waveform has the form of a linear ramp with a dc offset. A genetic algorithm was employed to find an optimized practically achievable waveform composed of a longer wavelength field, to provide the offset, in addition to higher frequency components. This second waveform is found to be capable of generating electron recollision energies as high as those for the perfect waveform while retaining the high recollision amplitudes of a pure sinusoidal wave. Calculations of high harmonic generation demonstrate this enhancement, by increasing the cutoff energy by a factor of 2.5 while maintaining the harmonic yield, providing an enhanced tool for attosecond science.

Nonlinear processes induced by the enhanced, evanescent field of surface plasmons excited by femtosecond laser pulses

Evanescent fields of surface plasmon polaritons (SPP) above metal surfaces can reach 1-2 orders of magnitude higher, nearly atomic field strengths in comparison to the relatively weak exciting laser fields of a femtosecond Ti:sapphire laser oscillator. We used these high plasmonic fields to study the characteristic SPP phenomena of intense field optics experimentally. It was found that both the intensity and the angular distribution of SPP emitted light depend nonlinearly on the exciting laser intensity in the higher-intensity, non-perturbative range of the interactions. These results are supported by our theory. At these strong excitations, an additional, depolarized, diffuse spectrum also appeared which can be attributed either to the fluorescence of Au, or to the non-equlibrium Planck radiation, originating from the fast cooling of the conduction electron cloud of Au excited by the femtosecond laser pulse.