Multiphoton Ionization Processes Research Articles

Sub-PPB Detection with Gas-Phase Multiphoton Electron Extraction Spectroscopy under Ambient Conditions.

Multiphoton electron extraction spectroscopy (MEES) is an advanced analytical technique that has demonstrated exceptional sensitivity and specificity for detecting molecular traces on solid and liquid surfaces. Building upon the solid-state MEES foundations, this study introduces the first application of MEES in the gas phase (gas-phase MEES), specifically designed for quantitative detection of gas traces at sub-part per billion (sub-PPB) concentrations under ambient atmospheric conditions. Our experimental setup utilizes resonant multiphoton ionization processes using ns laser pulses under a high electrical field. The generated photoelectron charges are recorded as a function of the laser's wavelength. This research showcases the high sensitivity of gas-phase MEES, achieving high spectral resolution with resonant peak widths less than 0.02 nm FWHM. We present results from quantitative analysis of benzene and aniline, two industrially and environmentally significant compounds, demonstrating linear responses in the sub-PPM and sub-PPB ranges. The enhanced sensitivity and resolution of gas-phase MEES offer a powerful approach to trace gas analysis, with potential applications in environmental monitoring, industrial safety, security screening, and medical diagnostics. This study confirms the advantages of gas-phase MEES over many traditional optical spectroscopic methods and demonstrates its potential in direct gas-trace sensing in ambient atmosphere.

Stark control of multiphoton ionization through Freeman resonances in alkyl iodides.

Multiphoton ionization (MPI) of alkyl iodides (RI, R = CnH2n+1, n = 1-4) has been investigated with femtosecond laser pulses centered at 800 and 400nm along with photoelectron imaging detection. In addition, the ultraviolet (UV)-vacuum ultraviolet (VUV) absorption spectra of gas-phase RIs have been measured in the photon energy range of 5-11eV using the VUV Fourier transform spectrometer at the VUV DESIRS beamline of the synchrotron SOLEIL facility. The use of high-laser-field strengths in matter-radiation interaction generates highly non-linear phenomena, such as the Stark shift effect, which distorts the potential energy surfaces of molecules by varying both the energy of electronic and rovibrational states and their ionization energies. The Stark shift can then generate resonances between intermediate states and an integer number of laser photons of a given wavelength, which are commonly known as Freeman resonances. Here, we study how the molecular structure of linear and branched alkyl iodides affects the UV-VUV absorption spectrum, the MPI process, and the generation of Freeman resonances. The obtained results reveal a dominant resonance in the experiments at 800nm, which counter-intuitively appears at the same photoelectron kinetic energy in the whole alkyl iodide series. The ionization pathways of this resonance strongly involve the 6p(2E3/2) Rydberg state with different degrees of vibrational excitation, revealing an energy compensation effect as the R-chain complexity increases.

Modulational instability and nonlinear dynamics of femtosecond lasers in transparent materials with non-Kerr nonlinearities

Femtosecond laser inscription in transparent materials is a physical process that finds widespread applications in material engineering, particularly in laser micromachining technology. In this process, the nonlinear optical response of the transparent material can be either intrinsic or induced by multiphoton ionization processes. In this work, a generic model is considered to describe the dynamics of femtosecond laser filamentation in transparent materials characterized by non-Kerr nonlinearities, focusing on the influence of multiphoton ionization processes in the generation of an electron plasma of inhomogeneous density. The mathematical model consists of a complex Ginzburg–Landau equation with a generalized saturable nonlinearity, besides the residual nonlinearity related to multiphoton ionization processes. This generalized complex Ginzburg–Landau equation is coupled to a rate equation for time evolution of the electron plasma density, where multiphoton ionizations are assumed to be the sole processes controlling the generation of the electron plasma. Dynamical properties of the model are discussed starting from the continuous-wave regime, where a modulational-instability analysis enables us to determine the stability conditions of continuous-wave modes in the system. The analysis reveals a dominant tendency of continuous-wave stability for relatively large values of the multiphoton ionization order K, provided the femtosecond laser operates in the anomalous dispersion regime. Numerical simulations of the mathematical model feature a family of wavetrains composed of self-focused, well-separated, pulse-shaped optical filaments whose repetition rates are shortened but amplitudes are increased, with an increase in K. Simulations suggest that such nonlinear wavetrain structures do not need the transparent material to be intrinsically nonlinear and that they may also be favored solely by the nonlinearity induced by multiphoton ionization processes in a linear transparent material.

Mapping Immune-Tumor Bidirectional Dialogue Using Ultrasensitive Nanosensors for Accurate Diagnosis of Lung Cancer.

Lung cancer is one of the most common cancers with high mortality worldwide despite the development of molecularly targeted therapies and immunotherapies. A significant challenge in managing lung cancer is the accurate diagnosis of cancerous lesions owing to the lack of sensitive and specific biomarkers. The current procedure necessitates an invasive tissue biopsy for diagnosis and molecular subtyping, which presents patients with risk, morbidity, anxiety, and high false-positive rates. The high-risk diagnostic approach has highlighted the need to search for a reliable, low-risk noninvasive diagnostic approach to capture lung cancer heterogeneity precisely. The immune interaction profile of lung cancer is driven by immune cells' distinctive, precise interactions with the tumor microenvironment. Here, we hypothesize that immune cells, particularly T cells, can be used for accurate lung cancer diagnosis by exploiting the distinctive immune-tumor interaction by detecting the immune-diagnostic signature. We have developed an ultrasensitive T-sense nanosensor to probe these specific diagnostic signatures using the physical synthesis process of multiphoton ionization. Our research employed predictive in vitro models of lung cancers, cancer-associated T cells (PCAT, MCAT) and CSC-associated T cells (PCSCAT, MCSCAT), from primary and metastatic lung cancer patients to reveal the immune-diagnostic signature and uncover the molecular, functional, and phenotypic separation between patient-derived T cells (PDT) and healthy samples. We demonstrated this by adopting a machine learning model trained with SERS data obtained using cocultured T cells with preclinical models (CAT, CSCAT) of primary (H69AR) and metastatic lung cancer (H1915). Interrogating these distinct signatures with PDT captured the complexity and diversity of the tumor-associated T cell signature across the patient population, exposing the clinical feasibility of immune diagnosis in an independent cohort of patient samples. Thus, our predictive approach using T cells from the patient peripheral blood showed a highly accurate diagnosis with a specificity and sensitivity of 94.1% and 100%, respectively, for primary lung cancer and 97.9% and 94.4% for metastatic lung cancer. Our results prove that the immune-diagnostic signature developed in this study could be used as a clinical technology for cancer diagnosis and determine the course of clinical management with T cells.

Soliton trains induced by femtosecond laser filamentations in transparent materials with saturable nonlinearity

Femtosecond laser inscriptions in optical media current offer the most reliable optical technology for processing of transparent materials, among which is the laser micromachining technology. In this process, the nonlinearity of the transparent medium can be either intrinsic or induced by multiphoton ionization processes. In this work, a generic model is proposed to describe the dynamics of femtosecond laser inscription in transparent materials characterized by a saturable nonlinearity. The model takes into account multiphoton ionization processes that can induce an electron plasma of inhomogeneous density and electron diffusions. The mathematical model is represented by a one-dimensional complex Ginzburg–Landau equation with a generalized saturable nonlinearity term in addition to the residual nonlinearity related to multiphoton ionization processes, coupled to a rate equation for time evolution of the electron plasma density. Dynamical properties of the model are investigated focusing on the nonlinear regime, where the model equations are transformed into a set of coupled first-order nonlinear ordinary differential equations, which are solved numerically with the help of a sixth-order Runge–Kutta algorithm with a fixed time step. Simulations reveal that upon propagation, spatiotemporal profiles of the optical field and of the plasma density are periodic pulse trains, the repetition rates and amplitudes of which are increased with an increase of both the multiphoton ionization order and the saturable nonlinearity. When electron diffusions are taken into account, the system dynamics remains qualitatively unchanged; however, the electron plasma density gets strongly depleted, leaving almost unchanged the amplitude of pulses composing the femtosecond laser soliton crystals.

Dissociative photoionization studies of ethyl iodide using synchrotron radiation photoionization mass spectrometry and photoelectron imaging

The stepwise dissociative photoionization of ethyl iodide(C2H5I) has been investigated via synchrotron radiation photoionization mass spectrometry and photoelectron imaging spectroscopy. By using VUV synchrotron radiation as excitation source, the photoionization and stepwise dissociative photoionization of C2H5I were observed by tuning the photon energy from 8.5 to 14.0 eV. The ionization energy of C2H5I was measured to be 9.33 ± 0.05 eV, and the appearance energies of C2H5+, C2H4+, C2H3+ were also determined according to the photoionization efficiency spectra. The possible pathways of the stepwise dissociation of C2H5I were determined by combining the experiments with theoretical calculation based on DFT. Finally, the multi-photon ionization process of C2H5I molecule was further studied via intensity-dependent photoelectron imaging spectroscopy.

Influence of low temperature on white light emission from graphene-based microchip

The laser-induced white emission of the graphene-based microchip was investigated under near-infrared region laser irradiation. The emission was characterized by an excitation threshold and an exponential dependence on the pump laser power. A decrease in temperature caused an increase in both the number of absorbed photons and the emission threshold. This dependence can be explained by the multiphoton ionization process in (sp2, sp3) hybridized domains.

Pushing multiphoton resonant ionization of the argon atom to the low-intensity regime

Resonance-enhanced multiphoton ionization process of the argon atom by an 800-nm 30-fs linearly polarized laser field is investigated at intensities range from 1.1 to $4.55\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$. At $4.55\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$ intensity the experimental photoelectron energy spectrum is in a good agreement with the time-dependent Schr\"odinger equation (TDSE) calculation where the double structure originating from dressed $4p\ensuremath{-}4d$ coupled transition is clearly identified. At lower intensity of $1.1\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$, the resonant ionization process via the $4f$ state is observed, however, the expected peak (jet) at ${90}^{\ensuremath{\circ}}$ in the photoelectron angular distribution from the zeroth order above threshold ionization, has vanished completely. Such behavior is attributed to the destructive interference phenomenon in the coherent contributions of different partial waves of the photoelectron, namely, $\ensuremath{\epsilon}d$ and $\ensuremath{\epsilon}g$ states, and has been confirmed in our TDSE calculations.

Nondipole effects and photoelectron momentum shifts in strong-field ionization by infrared light

High-order above-threshold ionization is studied theoretically beyond the electric dipole approximation. A hydrogen atom is assumed exposed to an ultrashort linearly polarized laser pulse with a central wavelength of 800 nm. The photoelectron momentum distribution from the corresponding nondipole multiphoton ionization process is calculated by means of numerical simulations with the time-dependent Schr\"odinger equation. In agreement with recent experiments as well as previous theoretical predictions, a forward-backward asymmetry with respect to the light propagation direction is revealed in the resulting momentum spectrum. Investigating separately the role of the linear and quadratic laser-matter interaction terms, i.e., the terms in the system Hamiltonian that are linear and quadratic in the laser's vector potential $\mathbit{A}$, respectively, it is found that the linear term introduces a shift of the electron momentum distribution in the laser propagation (forward) direction whereas the quadratic term is responsible for a corresponding backward shift. Nevertheless, only their net effect is observed in the final spectrum, and the quadratic contribution is shown to be responsible for the experimentally observed backward shift of the distribution for small electron momenta. Furthermore, for higher electron momenta it is found that the linear term takes over as the leading source of beyond-dipole ionization, effectively shifting the distribution in the forward direction.

Determination of atomic multiphoton-ionization phases by trichromatic multichannel wave-packet interferometry

We present the unambiguous determination of quantum phases in the multiphoton ionization (MPI) of potassium atoms by using a multichannel photoelectron interferometry scheme based on trichromatic pulse shaping. The colors of the laser field are chosen to produce three energetically separated photoelectron interferograms in the continuum. While the red pulse is two-photon resonant with the $3d$ state resulting in a $(2+1)$ resonance-enhanced MPI (REMPI), a $(1+2)$ REMPI occurs via the nonresonant intermediate $4p$ state with an initial green or blue pulse. We show that ionization via a nonresonant intermediate state lifts the degeneracy of photoelectron interferograms from pathways consisting of permutations of the colors. The analysis of the interferograms reveals a phase shift of $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ depending on the sign of the detunings in the $(1+2)$ REMPI pathways. In addition, we demonstrate that the photoionization time delay in the resonant $(2+1)$ REMPI pathway gives rise to a linear spectral phase in the photoelectron spectra. Insights into the underlying MPI processes are gained through an analytic perturbative description and numerical simulations of a trichromatic driven three-level system coupled to the continuum.

Transition from correlated to single-active-electron excitation in strontium nonlinear ionization

The nonlinear single and double ionization of Sr atoms as a result of their interaction with 800-nm Ti-sapphire laser pulses of 25-fs duration is experimentally investigated within the laser intensity range $I=3\phantom{\rule{0.16em}{0ex}}\mathrm{TW}\phantom{\rule{4pt}{0ex}}\mathrm{c}{\mathrm{m}}^{\text{--}2}--30\phantom{\rule{0.16em}{0ex}}\mathrm{TW}\phantom{\rule{4pt}{0ex}}\mathrm{c}{\mathrm{m}}^{\text{--}2}$. The latter range includes the over-the-barrier intensities for both Sr and ${\mathrm{Sr}}^{+}$, while the corresponding Keldysh adiabaticity parameters are larger than unity. Nevertheless, for $I\ensuremath{\ge}24\phantom{\rule{0.16em}{0ex}}\mathrm{TW}\phantom{\rule{4pt}{0ex}}\mathrm{c}{\mathrm{m}}^{\text{--}2}$ the recorded photoelectron-energy spectra show evidence of the (partial) applicability of tunneling or over-the-barrier ionization concepts, whereas below this value they can be discussed in terms of multiphoton ionization processes. More importantly, the data additionally point towards a further division of the multiphoton regime into two subranges separated roughly at $I\ensuremath{\sim}4\phantom{\rule{0.16em}{0ex}}\mathrm{TW}\phantom{\rule{4pt}{0ex}}\mathrm{c}{\mathrm{m}}^{\text{--}2}$. Above this value above-threshold-ionization structures and the single-active-electron picture dominate. On the contrary, at the lowest laser intensities the photoelectron spectra and angular distributions reveal single ionization pathways where the remaining ion is left into excited states. These pathways also require photon absorption in the continuum, which in the case of strontium and other alkaline-earth-metal atoms is structured by the presence of doubly excited states which are embedded in it. Hence, in this case ionization proceeds via quasiresonant ionization ladders formed by these doubly excited states which are dominated by the interaction between the two valence electrons of Sr. We discuss in detail the above rich phenomenology and relevant ionization mechanisms and propose possible directions of further work towards the elucidation of the role of configuration interaction in ionization processes.

Multichannel photoelectron phase lag across atomic barium autoionizing resonances

Phase lag associated with coherent control where an excited system decays into more than one product channel has been subjected to numerous investigations. Although previous theoretical studies have treated the phase lag across resonances in model calculations, quantitative agreement has never been achieved between the theoretical model and an experimental measurement of the phase lag from the $\ensuremath{\omega}\ensuremath{-}2\ensuremath{\omega}$ ionization of atomic barium. Yamazaki and Elliot, Phys. Rev. Lett. 98, 053001 (2007); Phys. Rev. A 76, 053401 (2007), suggesting that a toy model with phenomenological parameters is inadequate to describe the observed phase lag behavior. Here the phase lag is treated quantitatively in a multichannel coupling formulation, and our calculation based on a multichannel quantum defect and $R$-matrix treatment achieves good agreement with the experimental observations. Our treatment also develops formulas to describe the effects of hyperfine depolarization on multiphoton ionization processes. Moreover, we identify resonances between ${\text{Ba}}^{+}\phantom{\rule{4pt}{0ex}}5{d}_{3/2}$ and $5{d}_{5/2}$ thresholds that have apparently never been experimentally observed and classified.

Dissociation and ionization dynamics of CF3I and CH3I molecules via pump-and-probe experiments using soft x-ray free-electron laser

The dissociation and ionization dynamics of CF3I and CH3I molecules were investigated using a pump-and-probe technique that employs a soft x-ray free-electron laser (SACLA) in Japan. First, time-resolved inner-shell photoelectron spectroscopy was employed to observe the ultrafast reaction of CF3I by monitoring iodine 4d electrons. The change in the I 4d state observed in the photoelectron spectra is found to occur with a rise time τ of approximately 40 fs after a pump laser pulse, which is faster than that observed when an ultrafast gas-phase electron diffraction technique is employed. This implies that the inner-shell photoelectron spectroscopy is more sensitive to the potential surface near the Franck–Condon region. Second, a strong laser intensity at 266 nm, corresponding to a power density of 1.9 × 1014 W cm−2, can easily ionize CH3I molecules via multiphoton ionization processes, and the time dependence of the valence photoelectron spectra clearly shows that at the picosecond timescale, this pump laser pulse causes spectral peaks to shift owing to space-charge effects in response to the large amount of ions generated. Thus, the SACLA can be a useful tool to investigate not only the dynamical process of molecular dissociation but also the ionization process through the shift in the peaks of photoelectron spectra.

Photodissociation dynamics of methyl iodide across the A-band probed by femtosecond extreme ultraviolet photoelectron spectroscopy

The dissociation dynamics of CH3I at three UV pump wavelengths (279 nm, 254 nm, 243 nm) are measured using an extreme ultraviolet probe in a time-resolved photoelectron spectroscopy experiment. The results are compared with previously published data at a pump wavelength of 269 nm, [2020, Phys. Chem. Chem. Phys., 22, 25695], with complementary photoelectron spectroscopy experiments performed using a multiphoton ionization (MPI) probe [2019, Phys. Chem. Chem. Phys., 21, 11142] and with the recent action spectroscopy measurements of Murillo-Sánchez et al [2020, J. Chem. Phys., 152, 014304]. The measurements at 279 nm and 243 nm show signals that are consistent with rapid dissociation along the C–I bond occurring on timescales that are consistent with previous measurements. The measurements at 254 nm show a significantly longer excited state lifetime with a secondary feature appearing after 100 fs which is indicative of more complex dynamics in the excited state. The time-dependence of the changes are consistent with the previously measured MPI photoelectron spectroscopy measurements of Warne et al, [2019, Phys. Chem. Chem. Phys., 21, 11142]. The consistency of the signal appearance across ionization processes suggests that the extended observation time at 254 nm is not an artefact of the previously used MPI process but is caused by more complex dynamics on the excited state potential. Whether this is caused by complex vibrational dynamics on the dominant 3 Q 0 state or is due to enhanced population and dynamics on the 1 Q 1 state remains an open question.

The influence of molecular alignment and orientation in the ground state and excited state on the resonance-enhanced multi-photon ionization dynamics

We explore the influence of molecular alignment and orientation in the ground and excited states on the ionization probability, photoelectron angular distribution (PAD), energy-resolved photoelectron energy spectrum (PES) and two-dimensional momentum spectrum in the resonance-enhanced multiphoton ionization (REMPI) process. The calculated results for the LiH molecule show that molecular pre-alignment and -orientation have different effects on molecular photoionization. The ionization probability and energy-resolved photoelectron spectrum are associated with molecular pre-alignment. The angular distribution of photoelectrons and angular distribution of the momentum spectra are closely dependent on molecular pre-orientation. The ionization probability is also related to the center time and overlap area of the pump and probe pulses.

Numerical modeling of plasma formation in skin tissues induced by nanosecond pulsed laser

A numerical analysis is performed to investigate the comparative contribution of the mechanisms responsible for electron gain and losses in laser-induced breakdown of the skin tissues. In this regard, we adopted a simple theoretical formulation relying on the numerical solution of a rate equation that describes the growth of the electron density due to the joined effect of multiphoton, cascade and chromophore ionization processes. Here, the rate also includes the effect of electron loss due to diffusion and recombination processes. The analysis considered skin tissue irradiated by a Nd:YAG laser radiation in the 200–550 nm wavelength range with 6 ns pulse duration full-width half-maximum (FWHM).

Picosecond laser ablation on glass using wavelength of 1064, 532, and 355 nm

The process of removing transparent material such as glass by means of laser ablation is widely performed in the industry. In many cases, the ultrashort pulse laser is used as the laser source, but the laser wavelength is diverse. The ablation mechanism of the transparent material generally is considered nonlinear absorption, but it is known that the absorption coefficient depends on the wavelength. However, there are few examples that discuss the ablation mechanism for multiwavelengths. We performed laser ablation of glass with a 15-ps pulse width and 1064, 532, and 355 nm wavelengths and constructed a unified understanding of the principle by numerical calculation using combined rate equation and beam propagation method. In addition, we measured the three-dimensional profile of the processing traces using a microscope and investigated the dependence of ablation volume to the wavelength. As a result, we acquired the results of ablation efficiency, laser fluence threshold, and inverse absorption coefficient of fused silica glass using various wavelengths in a single set-up. We were able to explain the ablation diameter and depth by numerical calculation considering the multiphoton ionization process and avalanche ionization process. In addition, the calculated laser efficiency indicated that the wavelength suitable for ablation on glass was 355 nm.

Photoelectron imaging on vibrational excitation and Rydberg intermediate states in multi-photon ionization process of NH3 molecule**Project supported by the National Natural Science Foundation of China (Grant Nos. 11574116, 11534004, and 10704028).

The ionization processes of NH3 molecule are studied by photoelectron velocity map imaging technique in a linearly polarized 400-nm femtosecond laser field. The two-dimensional photoelectron images from ammonia molecules under different laser intensities are obtained. In the slow electron region, the values of kinetic energy of photoelectrons corresponding to peaks 1, 2, 3, and 4 are 0.27, 0.86, 1.16, and 1.6 eV, respectively. With both the kinetic energy and angular distribution of photoelectrons from NH3 molecules, we can confirm that the two-photon excited intermediate Rydberg state is A∼1 () state for photoelectron peaks 2, 3, 4, and the three peaks are marked as 1223 (2 + 2), 1123 (2 + 2), and 1023 (2 + 2) multi-photon processes, respectively. Then, peak 1 is found by adding a hexapole between the source chamber and the detection chamber to realize the rotational state selection and beam focusing. Peak 1 is labeled as the 1323 (3 + 1) multi-photon process through the intermediate Rydberg state . The phenomena of channel switching are found in the slow electron kinetic energy distributions. Our calculations and experimental results indicate that the stretching vibrational mode of ammonia molecules varies with channels, while the umbrella vibration does not. In addition, we consider and discuss the ac-Stark effect in a strong laser field. Peaks 5 and 6 are marked as (2 + 2 + 1) and (2 + 2 + 2) above threshold ionization processes in the fast electron region.

Numerical investigation of the plasma formation in air generated by 355 nm Nd:YAG laser pulses

In the present work, a numerical analysis is performed to investigate the comparative contribution of the mechanisms responsible for electron gain and losses in laser-induced breakdown. In this regard, we adopted a simple theoretical formulation relying on the numerical solution of a rate equation that describes the growth of the electron density due to the joined effect of multiphoton and cascade ionization processes. The rate equation also includes the effect of electron loss due to diffusion, attachment and recombination processes. The analysis considered atmospheric air irradiated by a Nd:YAG laser radiation at a wavelength of 355 nm with 5 ns pulse duration full-width half-maximum (FWHM).

ЧИСЕЛЬНИЙ РОЗВ'ЯЗОК КРАЙОВОЇ ЗАДАЧІ ДЛЯ ПАРАБОЛІЧНОГО РІВНЯННЯ З НЕСАМОСПРЯЖЕНИМ ОПЕРАТОРОМ І ЗВ'ЯЗАНИМИ КРАЙОВИМИ УМОВАМИ

A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.