Femtosecond pulse propagation in air: variational analysis

Simulation of spatiotemporal light dynamics based on the time-dependent Schrödinger equation

We establish a first-principle model for the simulation of spatiotemporal light pulse dynamics based on the combination of the time-dependent Schrödinger equation and the unidirectional propagation equation. The proposed numerical scheme enables computationally efficient simulation while being stable and accurate. We use the new model to examine self-focusing of a short pulse in atomic hydrogen and show that an accurate description of the excited-levels dynamics can only be achieved by a propagation model with an ab-initio description of the light-matter interaction, which accounts for the laser-dressed multilevel structure of the system, including bound and free states, and its sub-cycle response.

Experimental demonstration of dyed water aerosol fluorescence stimulated by femtosecond laser postfilaments propagating in air

We present the fluorescence spectra of single millimeter water droplets and micron-sized dyed water aerosol (rhodamine 6G) stimulated by a high-intensity femtosecond Ti:sapphire-laser pulse (carrier wavelength 792 nm) upon its nonlinear propagation in air. The distinctive feature of our experimental measurements is that the droplet fluorescence is obtained in the area of plasma-free pulse propagation after the pulse filamentation has already been terminated (postfilamentation region). Our results significantly expand the working area of femtosecond laser-induced fluorescence spectroscopy for remote diagnostics of atmospheric aerosols.

Revealing Local Temporal Profile of Laser Pulses of Intensity above 1014 W/cm2

We demonstrated a method for in situ temporal characterization of an intense femtosecond laser pulse around its focus where the laser intensity exceeds 10¹⁴ W/cm². Our method is based on the second harmonic generation (SHG) by a relatively weak femtosecond probe pulse and the intense femtosecond pulses under analysis in the gas plasma. With the increase in the gas pressure, it was found that the incident pulse evolves from a Gaussian profile to a more complicated structure featured by multiple peaks in the temporal domain. Numerical simulations of filamentation propagation support the experimental observations of temporal evolution. This simple method can be applied to many situations involving femtosecond laser-gas interaction, when the temporal profile of the femtosecond pump laser pulse with an intensity above 10¹⁴ W/cm² cannot be measured in traditional ways.

Simulation of nonlinear propagation of femtosecond laser pulses in air for quantitative prediction of the ablation crater shape

The utilization of sub-100 fs pulses has attracted attention as an approach to further improve the quality and precision of femtosecond laser microfabrication. However, when using such lasers at pulse energies typical for laser processing, nonlinear propagation effects in air are known to distort the beam's temporal and spatial intensity profile. Due to this distortion, it has been difficult to quantitatively predict the final processed crater shape of materials ablated by such lasers. In this study, we developed a method to quantitatively predict the ablation crater shape, utilizing nonlinear propagation simulations. Investigations revealed that the ablation crater diameters derived by our method were in excellent quantitative agreement with experimental results for several metals over a two-orders-of-magnitude range in the pulse energy. We also found a good quantitative correlation between the simulated central fluence and the ablation depth. Such methods should improve the controllability of laser processing with sub-100 fs pulses and contribute to furthering their practical application to processes over a wide pulse-energy range, including conditions with nonlinear-propagating pulses.

Observation of two-times self-focusing of femtosecond laser beam in ZnO crystal by two-photon luminescence

By "seeing" the green two-photon luminescence, two separate focusing points are observed on the propagation axis of a converging femtosecond laser beam in a ZnO single crystal rod. It is found that the self-focusing effect makes a significant contribution to the formation of the first focusing point, while the second focusing point is caused by self-refocusing. The position of the first focusing point is in good agreement with the value predicted by a model developed by Chin and his co-workers. These experimental findings could be the unprecedented evidence for the self-focusing and refocusing effect of the femtosecond laser filament propagation in nonlinear media.

Modeling and simulation techniques in extreme nonlinear optics of gaseous and condensed media

Computer simulation techniques for extreme nonlinear optics are reviewed with emphasis on the high light-intensity regimes in which both bound and freed electronic states contribute to the medium response and thus affect the optical pulse dynamics. The first part concentrates on the optical pulse propagation modeling, and provides a classification of various approaches to optical-field evolution equations. Light-matter interaction models are reviewed in the second part, which concentrates on methods that can be integrated with time- and space-resolved simulations encompassing realistic experimental scenarios.

Pattern dynamics and filamentation of femtosecond terawatt laser pulses in air including the higher-order Kerr effects

Plasma defocusing and higher-order Kerr effects on multiple filamentation and pattern formation of ultrashort laser pulse propagation in air are investigated. Linear analyses and numerical results show that these two saturable nonlinear effects can destroy the coherent evolution of the laser field, and small-scale spatial turbulent structures rapidly appear. For the two-dimensional case, numerical simulations show that blow-up-like solutions, spatial chaos, and pseudorecurrence can appear at higher laser intensities if only plasma defocusing is included. These complex patterns result from the stochastic evolution of the higher- or shorter-wavelength modes of the laser light spectrum. From the viewpoint of nonlinear dynamics, filamentation can be attributed to the modulational instability of these spatial incoherent localized structures. Furthermore, filament patterns associated with multiphoton ionization of the air molecules with and without higher-order Kerr effects are compared.

High-intensive femtosecond singular pulses in Kerr dielectrics

The nonlinear dynamics of a high-power femtosecond singular pulse in Kerr media are analyzed numerically upon optically induced ionization. We examine the plasma inertia impact to stable propagation of optical vortices. Multifoci behavior of vortices in medium are revealed. Next we numerically demonstrate that inertial character of plasma formation provides a quasi-soliton regime of vortex propagation resistant to symmetry-breaking perturbation.

Self-focusing arrest of femtosecond laser pulses in air at different pressures

We study analytically and numerically the self-focusing arrest of femtosecond laser pulses in air at different pressures in the presence of an external focus lens. Analytical estimations as well as results of simulations show that the intensity at which the self-focusing arrest occurs is almost independent of the gas pressure. However, a dependence on the temperature is found. The Raman effect is taken into account, and an estimation of the intensity inside of filaments at high altitudes is given.

Variational analysis of self-focusing of intense ultrashort pulses in gases

By using perturbation theory we derive an expression for the electrical field of a Gaussian laser beam propagating in a gas medium. This expression is used as a trial solution in a variational method to get quasi-analytical solutions for the width, intensity, and self-focusing distance. The approximation gives a better agreement with results of numerical simulations for a broad range of values of the input power than previous analytical results available in the literature. The results apply in the case of ultrashort pulses too.

Stability and transient effects in nanosecond ultraviolet light filaments in air

We investigate the transient behavior and stability of nanosecond duration ultraviolet pulses propagating in air. Both the transient behavior arising from the finite pulse duration and the modulational instability, are found to cause pulses to fragment over lengths on the scale of meters. We discuss the theoretical and experimental implications of the instability and transient effects for long duration pulse propagating in air and generating filaments. In particular, our results imply that continuous-wave models are very limited when used to predict dynamical properties of pulse propagation.

Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization

Propagation of an optical high-power cylindrically symmetric beam in a material characterized by cubic-quintic nonlinearity is studied both analytically and numerically. In this case we have to consider the self-defocusing effect caused by the presence of free electrons produced due to plasma formation. The variational method is used to study the system analytically. The finite-difference beam propagation method is used for the numerical analysis. Stable (2+1) D spatial solitons are observed. The analytical results are found to be in very good agreement with the numerical results.

Long-range detection and length estimation of light filaments using extra-attenuation of terawatt femtosecond laser pulses propagating in air

High-energy femtosecond laser pulses propagating in the atmosphere undergo self-focusing resulting in the appearance of the phenomenon of filamentation. We observed an extra-attenuation of such (terawatt) femtosecond laser pulses propagating in the atmosphere when compared with long pulses (200 ps) with the same energy. This is because, in contrast to the linear propagation of the long pulse, the input femtosecond laser pulse is attenuated owing to either absorption through multiphoton ionization/tunnel ionization or to scattering on the laser-induced plasma; self-phase-modulation and self-steepening further convert partially the energy initially contained in the fundamental bandwidth into the broad side bands of the laser, becoming eventually a white-light laser pulse (supercontinuum). The experimental data allow us to extract an effective extra-attenuation coefficient for an exponential decay of the input pulse energy with the propagation distance. Such a coefficient allows us to estimate an upper bound of the filament length under the experimental conditions used. More generally, our observation leads to a new technique to remotely detect light filaments in the atmosphere.

Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air

The long-range propagation of two-colored femtosecond filaments produced by an infrared (IR) ultrashort pulse exciting third harmonics (TH) in the atmosphere is investigated, both theoretically and experimentally. First, it is shown that the coupling between the pump and TH components is responsible for a wide spectral broadening, extending from ultraviolet (UV) wavelengths (220 nm) to the mid-IR (4.5 microm). Supercontinuum generation takes place continuously as the laser beam propagates, while TH emission occurs with a conversion efficiency as high as 0.5%. Second, the TH pulse is proven to stabilize the IR filament like a saturable quintic nonlinearity through four-wave mixing and cross-phase modulation. Third, the filamentation is accompanied by a conical emission of the beam, which becomes enlarged at UV wavelengths. These properties are revealed by numerical simulations and direct experimental observations performed from the Teramobile laser facility.

Nonlinear optical pulse propagation simulation: from Maxwell's to unidirectional equations

Spatial- and time-domain versions of the unidirectional pulse propagation equation (UPPE) are derived and compared from the point of view of their practical application in simulations of nonlinear optical pulse dynamics. A modification of the UPPE suitable for ultrathin optical waveguides, such as submicron silica wires, is also presented. We show in detail how various, previously published propagation equations follow from the UPPE in a unified way that clearly elucidates their underlying approximations and areas of applicability.

Time-resolved measurements of self-focusing pulses in air

The spatial, spectral, and temporal properties of self-focusing 798-nm 100-fs pulses in air are experimentally measured with high-resolution, single-shot techniques at a set propagation distance of 10.91 m. The data, obtained with an initially collimated Gaussian beam, show significant evolution of spatial narrowing as well as temporal and spectral changes at intensities lower than those required for significant ionization of air.

White-light filaments for atmospheric analysis

Most long-path remote spectroscopic studies of the atmosphere rely on ambient light or narrow-band lasers. High-power femtosecond laser pulses have been found to propagate in the atmosphere as dynamically self-guided filaments that emit in a continuum from the ultraviolet to the infrared. This white light exhibits a directional behavior with enhanced backward scattering and was detected from an altitude of more than 20 kilometers. This light source opens the way to white-light and nonlinear light detection and ranging applications for atmospheric trace-gas remote sensing or remote identification of aerosols. Air ionization inside the filaments also opens promising perspectives for laser-induced condensation and lightning control. The mobile femtosecond-terawatt laser system, Teramobile, has been constructed to study these applications.

Propagation of intense short laser pulses in the atmosphere

The propagation of short, intense laser pulses in the atmosphere is investigated theoretically and numerically. A set of three-dimensional (3D), nonlinear propagation equations is derived, which includes the effects of dispersion, nonlinear self-focusing, stimulated molecular Raman scattering, multiphoton and tunneling ionization, energy depletion due to ionization, relativistic focusing, and ponderomotively excited plasma wakefields. The instantaneous frequency spread along a laser pulse in air, which develops due to various nonlinear effects, is analyzed and discussed. Coupled equations for the power, spot size, and electron density are derived for an intense ionizing laser pulse. From these equations we obtain an equilibrium for a single optical-plasma filament, which involves a balancing between diffraction, nonlinear self-focusing, and plasma defocusing. The equilibrium is shown to require a specific distribution of power along the filament. It is found that in the presence of ionization a self-guided optical filament is not realizable. A method for generating a remote spark in the atmosphere is proposed, which utilizes the dispersive and nonlinear properties of air to cause a low-intensity chirped laser pulse to compress both longitudinally and transversely. For optimally chosen parameters, we find that the transverse and longitudinal focal lengths can be made to coincide, resulting in rapid intensity increase, ionization, and white light generation in a localized region far from the source. Coupled equations for the laser spot size and pulse duration are derived, which can describe the focusing and compression process in the low-intensity regime. More general examples involving beam focusing, compression, ionization, and white light generation near the focal region are studied by numerically solving the full set of 3D, nonlinear propagation equations.

Femtosecond snapshot imaging of propagating light itself

An ultrafast imaging technique has been developed to visualize directly a light pulse that is propagating in a medium. The method, called femtosecond time-resolved optical polarigraphy (FTOP), senses instantaneous changes in the birefringence within the medium that are induced by the propagation of an intense light. A snapshot sequence composed of each femtosecond probing the pulse delay enables ultrafast propagation dynamics of the intense femtosecond laser pulse in the medium, such as gases and liquids, to be visualized directly. Other examples include the filamentation dynamics in CS2 liquid and the propagation dynamics in air related to the interaction with laser breakdown plasma. FTOP can also be used to extract information on the optical Kerr constant and its decay time in media. This method is useful in the monitoring of the intensity distribution in the nonlinear propagation of intense light pulses, which is a frequently studied subject in the field of physics regarding nonlinear optics and laser processing.

Multiple foci and a long filament observed with focused femtosecond pulse propagation in fused silica

Multiple foci and a long filament are observed when we focus a femtosecond laser pulse into a fused-silica sample. The dependences of the intensity distribution of the plasma luminescence on the pulse energy and the numerical aperture (NA) of the focusing objective are investigated. Multiple foci are observed when NA of </=0.65 . A long filament tail is formed instead of multiple foci when the NA is 0.85. A physical image of femtoseond pulse propagation is given by a model based on the nonlinear Schrödinger equation.

Multiframe observation of an intense femtosecond optical pulse propagating in air

We have demonstrated femtosecond time-resolved and picosecond time-interval successive observations of a single femtosecond optical pulse propagating in air with ultrafast self-modulation such as filamentation. A quadruple femtosecond probe pulse crossing an intense propagating pulse at picosecond intervals was able to capture directly four successive images of the propagating pulse as in a movie but with femtosecond time resolution. From this observation, we can directly analyze and recognize the propagation process, which is significantly affected by pulse-energy fluctuation, and (or) atmospheric turbulence from shot to shot.

Intense pulses in air: breakup of rotational symmetry

We investigate azimuthal instabilities of intense rotationally symmetric pulsed beams propagating in air. Although the spatial-temporal evolution of the field is strongly influenced by the onset of plasma generation, the instabilities are caused chiefly by the Kerr effect. We conclude that calculations that assume rotational symmetry become unrealistic because of the fast growth of azimuthal instabilities shortly after the onset of plasma generation.

Polarization dynamics of femtosecond pulses propagating in air

Polarization dynamics of femtosecond light pulses propagating in air is studied by computer simulation. A rich variety of dynamics is found that depends on the initial polarization state and power of the pulse. Effects of polarization on the plasma and supercontinuum generation are also discussed.

Backward supercontinuum emission from a filament generated by ultrashort laser pulses in air

Backward emission of the supercontinuum from a light filament induced by high-intensity femtosecond laser pulses propagating in air has been observed to be enhanced compared with linear Rayleigh-Mie scattering. This enhancement is interpreted as a nonlinear scattering process onto longitudinal refractive-index changes induced by the laser pulse itself. The spectral dependence of the supercontinuum angular distribution is also investigated.