Nonlinear Response of Diluted Gases to an Ultraviolet Femtosecond Pulse

Quantum-mechanical simulations of the nonlinear response of a one-dimensional quantum system with the energy structure close to that of the xenon atom to an ultraviolet femtosecond pulse with an intensity of 1–100 TW/cm² reveal the dispersion of the cubic nonlinearity coefficient in the range of 266–400 nm and its intensity dependence. This excludes the description of the response of bound electrons as ${{\chi }^{{(3)}}}{{E}^{3}}$. The calculation of the polarization with this one-dimensional quantum model can be used to simulate the propagation of ultraviolet femtosecond radiation in a gas.

1 INTRODUCTION

To date, a detailed experimental and theoretical study of the propagation and filamentation of intense infrared femtosecond laser radiation in the volume of a gas [1] and in a gas-filled fiber [2] has been carried out. The nonlinear response of gaseous media to an infrared femtosecond pulse can be described theoretically using a phenomenological approach [3], which represents nonlinearity as a superposition of the current of electrons released in multiphoton/tunnel ionization events [4] and the time derivative of the nonlinear polarization of the medium P_nl, corresponding to the anharmonic motion of bound electrons. The polarization P_nl is mainly determined by the instantaneous contribution ${{P}_{{{\text{nl}}}}} = {{\chi }^{{(3)}}}{{E}^{3}}$ cubic in the field E [5–8], where the third-order nonlinearity coefficient ${{n}_{2}} \propto {{\chi }^{{(3)}}}$ is found from experiments [9–11]. To simulate the pulse propagation in molecular gases, the model is provided with a description of the resonant linear [11–14] and inertial cubic [14–18] responses.

The phenomenological approach to the description of nonlinearity is not justified mathematically. Its applicability is due to a significantly different number of photons that provide the response of bound electrons ($3\hbar \omega$) and ionization of the medium ($8\hbar \omega$ for the  О₂ molecule at the central laser wavelength $\lambda =$ 2πc/ω = 800 nm, where c is the speed of light). For femtosecond pulses with the central wavelength in the near and mid-infrared ranges, the phenomenological approach reproduces the experimental results. For example, for a titanium–sapphire laser with a central wavelength of about 800 nm, the peak intensity in a filament in air estimated from measurements and simulations is about 100 TW/cm² [19–22].

In the plasma channel of an ultraviolet filament, radio-frequency and terahertz radiation can be amplified owing to the relatively narrow spectrum of photoelectrons [23]. However, the parameters of ultraviolet filaments are studied less than those for infrared radiation. In particular, even the order of the pulse intensity at a wavelength of 250 nm (the third harmonic of a titanium-sapphire laser) in a filament is unknown: according to [24], it is about 0.1 TW/cm², while according to [25], it is 20 TW/cm². The very possibility of applying the phenomenological approach [25–28] to the theoretical description of the interaction of high-intensity ultraviolet femtosecond laser radiation with gaseous media is doubtful. Indeed, the ionization of oxygen or xenon by 250-nm radiation becomes three-photon [29]. Therefore, the response of electrons bound in atoms or molecules and released in ionization events [4, 30] becomes cubic in the field.

The limited applicability of the phenomenological approach to the description of the gas nonlinearity in the ultraviolet range was indirectly confirmed in [31], where the dependence of the coefficient ${{n}_{2}}$, obtained from quantum simulation, on the wavelength λ was approximated by a Sellmeier-type formula. A singularity in the approximation ${{n}_{2}}(\lambda )$ was at a wavelength corresponding to one-third of the ionization potential, i.e., when nonlinear ionization becomes three-photon. Thus, for the femtosecond ultraviolet radiation with a frequency near one-third of the ionization potential, the very definition ${{n}_{2}}$ may lose its physical sense.

It should be noted that the quantum mechanical description of the nonlinear response of semiconductors and dielectrics is usually based on the system of Maxwell–Bloch equations, see, e.g., [32, 33]. In the case of a diluted monatomic gas, there is no need to take into account the band structure, anisotropy of the medium, interaction with the reservoir, and plasma thermalization, so the time-dependent Schrödinger equation [34, 35] can be used instead.

In this work, we use a one-dimensional quantum-mechanical model [36] of the interaction of light with a potential well with energy levels approximately corresponding to the ground and excited states of a xenon atom to study the nonlinear response induced in a medium by a femtosecond pulse. In a wide range of pulse intensities from 0.1 to 100 TW/cm² and its central wavelengths from 266 to 1500 nm, we determine the nonlinear response of a medium consisting of such noninteracting quantum systems from the numerical solution of the time-dependent Schrödinger equation. The simulated nonlinear polarization in the infrared range corresponds to the response determined from the phenomenological approach. For ultraviolet femtosecond pulses, this agreement is violated: even at a low intensity of 2 TW/cm², the nonlinear response cannot be approximated by the cube of the field. As the intensity increases to 25 TW/cm², the temporal dependences of the nonlinear polarization simulated quantum-mechanically and phenomenologically oscillate with a phase shift of a quarter of the optical period and have different frequency spectra.

2 ONE-DIMENSIONAL QUANTUM MECHANICAL MODEL OF THE INTERACTION OF A FEMTOSECOND PULSE WITH GAS

Let $U(x)$ be a one-dimensional potential well. To determine the bound states of the potential $U(x)$, we used an iterative algorithm, which is described in detail in the Appendix. By varying the parameters, we have built the potential (hereinafter, atomic units are used in the formulas)

$$U(x) = - \frac{{0.5625}}{{\sqrt {{{x}^{2}} + {{{0.63}}^{2}}} }}\exp \left[ { - {{{\left( {\frac{x}{8}} \right)}}^{{16}}}} \right],$$

(1)

where the ${{W}_{0}} = - 12.08$, ${{W}_{1}} = - 2.93$, and W₂ = –1.17 bound states ${\text{|}}{{\Psi }_{j}}\rangle$, $j = 0,1,2$, have the energies W₀ = −12.08 eV, W₁ = −2.93 eV, and W₂ = −1.17 eV reproducing the energy structure of the xenon atom (see Fig. 1).

Fig. 1.

(Color online) (a) (Black line) Potential $U(x)$ and (color lines) wavefunctions of bound states ${{\Psi }_{j}}(x)$. (b) Atomic levels of xenon.

To describe the interaction of light with this system, we used the time-dependent Schrödinger equation for the wavefunction $\Psi (x,t)$ with the initial condition $\Psi (x,t \to - \infty ) = {\text{|}}{{\Psi }_{0}}\rangle$:

$$i\frac{{\partial \Psi }}{{\partial t}} = - \frac{1}{2}\frac{{{{\partial }^{2}}\Psi }}{{\partial {{x}^{2}}}} + U(x)\Psi - E(t)x\Psi ,$$

(2)

where $E(t) = \sqrt {{{I}_{0}}} \exp ( - {{t}^{2}}{\text{/}}[2\tau _{0}^{2}])\sin {{\omega }_{0}}t$ is the electric field of a laser pulse with the duration $2{{\tau }_{0}} = 10$ fs. The radiation intensity ${{I}_{0}}$ varied from 0.1 to 100 TW/cm² (from 3 × 10⁻⁶ to 3 × 10⁻³ atomic intensity); the frequency ${{\omega }_{0}}$ corresponded to wavelengths from 1500 to 266 nm, i.e., from the low-frequency tunnel limit to the three-photon ionization.

The simulations were carried out on an NVIDIA GeForce RTX 3080 video card using the CUDA technology. The temporal domain was $20{\kern 1pt} {{\tau }_{0}} = 100$ fs, and the spatial domain was 8192 a.u. The number of nodes in time and space grids $N{{ = 2}^{{16}}}$ provided the time resolution $\Delta t = 1.5$ as and the space resolution $\Delta x = 0.125$ a.u. The typical calculation time was about 5 min.

The ionization probability $\eta = 1 - \sum\nolimits_j {\kern 1pt} {\text{|}}\langle {{\Psi }_{j}}\Psi \rangle {{{\text{|}}}^{2}}$ obtained by the numerical integration of Eq. (2) is in agreement with the results provided by the Perelomov–Popov–Terent’ev formula. For the central wavelengths of 266, 800 and 1500 nm, the dependences of the probability of the transition of the system from a bound state to a free one on the peak pulse intensity are shown in Fig. 2. The dependences $\eta ({{I}_{0}})$ obtained from quantum mechanical simulations correspond to the multiphoton limit for ultraviolet pulses and the tunnel limit for near and mid-infrared pulses.

Fig. 2.

(Color online) (Points) Calculated ionization probability in a wide range of peak intensities of a 10-fs pulse and (lines) its approximation in the multiphoton and tunnel limits.

3 NONLINEAR RESPONSE OF A ONE-DIMENSIONAL QUANTUM SYSTEM

Since we consider diluted gaseous media, the macroscopic polarization P is the product of the dipole moment of the atom and the concentration of a gas. We normalize the macroscopic polarization to the concentration. The normalized polarization is equal to the dipole moment of our quantum system:

$$P(t) = - \langle \Psi {\text{|}}\hat {x}{\text{|}}\Psi \rangle .$$

(3)

The polarization $P(t)$ contains linear and nonlinear contributions of bound electrons, as well as the contribution of free electrons and interference terms, which cannot be separated in the general case. We determine the effective cubic nonlinear coefficient ${{n}_{2}} \propto {{\chi }^{{(3)}}}$ from the best approximation of the polarization given by Eq. (3) by the dependence

$$P(t) = {{\chi }^{{(1)}}}(t) \otimes E(t) + {{\chi }^{{(3)}}}{{E}^{3}}(t),$$

(4)

where the sign $\otimes$ denotes convolution. The nonlinear part of the polarization can be distinguished using simulations at several peak intensities ${{I}_{0}}$. In this case, the dispersion of the linear response of the gaseous medium is taken into account for the whole peak intensity range.

The effective cubic nonlinear coefficient ${{n}_{2}}$ found from the approximation given by Eq. (4) is shown in Fig. 3 as a function of the intensity for three wavelengths $\lambda = 1500,$ 800, and 266 nm. These ${{n}_{2}}$ values are approximately an order of magnitude lower than the Kerr nonlinear coefficient of gases in the infrared range ${{n}_{2}} \sim {{10}^{{ - 19}}}$ cm²/W known from experiments [9–11]. For pulses with $\lambda = 1500$ and 800 nm, the effective coefficient ${{n}_{2}}$ is constant up to intensities of 30–40 TW/cm², after which it becomes negative because of a significant fraction of ionized atoms ($\eta \gtrsim {{10}^{{ - 3}}}$, see Fig. 2). In the case $\lambda = 266$ nm, the ionization probability increases “gradually” with the intensity, and the ${{n}_{2}}$ coefficient cannot be considered as constant already at intensities higher than 5 TW/cm².

Fig. 3.

(Color online) Effective cubic nonlinearity coefficient versus the intensity at three wavelengths. For infrared pulses, the dependence can be considered constant; for ultraviolet pulses, it decreases linearly (lines).

To demonstrate the cubic nonlinearity dispersion, the dependence of the effective coefficient ${{n}_{2}}$ on the central frequency ω₀ of the laser pulse is shown in Fig. 4. In accordance with the simulations [31], it can be approximated by a Sellmeier-type formula [26] if the resonant frequencies are one-third of the resonant frequencies of the linear dispersion, the latter being the frequencies of ground-to-continuum and ground-to-excited transitions:

$${{n}_{2}}({{\omega }_{0}}) = \frac{A}{{\Omega _{A}^{2} - \omega _{0}^{2}}} + \frac{B}{{\Omega _{B}^{2} - \omega _{0}^{2}}},$$

(5)

where ${{\Omega }_{A}} = {\text{|}}{{W}_{0}}{\text{|/}}3$ = 4.03 eV and ${{\Omega }_{B}} = {\text{|}}{{W}_{0}} - {{W}_{1}}{\text{|/}}3$ = 3.05 eV. At frequencies above 600–700 THz (wavelengths less than 430–500 nm), the ${{n}_{2}}$ values change significantly on a scale of the femtosecond pulse spectral width and become negative. The approximation given by Eq. (4) is no longer physically meaningful since the third-order nonlinear response is delayed and/or depends on the radiation intensity.

Fig. 4.

(Color online) (Circles) Effective cubic nonlinearity coefficient ${{n}_{2}}$ estimated by Eq. (4) from quantum mechanical simulations with Eq. (2) versus the central frequency of a femtosecond pulse with a peak intensity of 10 TW/cm² and (red line) its approximation by the Sellmeier-type formula (5).

For an infrared pulse (1500 nm, Fig. 5), the nonlinear polarization of a quantum system can be reproduced with a high accuracy using a phenomenological model with the appropriately chosen cubic nonlinear coefficient and the pre-exponential factor in the ionization rate. High-frequency polarization oscillations at the trailing edge of the pulse correspond to the recombination generation of harmonics [37, 38]. For an ultraviolet pulse (266 nm, Fig. 6), one can match the amplitudes of $P(t)$ calculated in two approaches, but at the same time, they have a significantly different phase, intra-period dynamics, and frequency spectrum (cf. Figs. 7a and 7b). Even at a low intensity of 2 TW/cm² (see Fig. 6a), when the fraction of ionized atoms is $\eta \sim {{10}^{{ - 5}}}$, the polarization in the quantum system is not proportional to the cube of the electric field but lags behind it. With the increase in the intensity to 25 TW/cm², the delay of the nonlinear response found numerically by solving the Schrödinger equation with respect to the response calculated using the phenomenological approach increases, reaching a quarter of the optical period (see Figs. 6b, 6c).

Fig. 5.

(Color online) Nonlinear polarization of an atomic system obtained in (red lines) quantum mechanical simulations and (black lines) the phenomenological model for pulses at a wavelength of 1500 nm with an intensity of (a) 15 and (b, c) 59 TW/cm². (d) Laser pulse field.

Fig. 6.

(Color online) Same as in Fig. 5, but for pulses at a wavelength of 266 nm with an intensity of (a) 2 and (b, c) 25 TW/cm².

Fig. 7.

(Color online) Nonlinear polarization spectra for (a) infrared and (b) ultraviolet pulses obtained from the dependences plotted in Figs. 5c and 6c, respectively. The gray vertical lines indicate the central frequencies of the pulses.

4 CONCLUSIONS

The nonlinear response of an atom to a femtosecond pulse with central wavelengths from 266 to 1500 nm has been studied using the developed one-dimensional quantum mechanical model of the interaction of laser radiation with gas. In the ultraviolet part of the spectrum, the effective cubic nonlinear coefficient exhibits dispersion and a strong dependence on the intensity; i.e., ${{\chi }^{{(3)}}}$ cannot be considered constant. Even at an intensity of ~1 TW/cm², the nonlinear response to an ultraviolet pulse is not cubic in the field. With the increase in the intensity to ~20 TW/cm², the deviation of the results of the quantum mechanical simulations of the nonlinear polarization from the results of the simulations using the phenomenological approach increases.

Thus, quantum mechanical simulations of the nonlinear response of an atomic system to an ultraviolet femtosecond pulse are apparently necessary for the description of the propagation of such pulses in a gaseous medium since phenomenological models become inapplicable when the response of bound electrons and ionization involve the same number of photons. The numerical cost of quantum simulations based on the one-dimensional model allows the use of this approach to simulate the propagation of ultraviolet femtosecond radiation in a gas.

ITERATIVE ALGORITHM FOR DETERMINING THE BOUND STATES OF THE POTENTIAL WELL

To determine the energy levels of the given potential $U(x)$ and wavefunctions ${\text{|}}{{\Psi }_{j}}\rangle$, the operator $(1 - \alpha \hat {H})$ is iteratively applied to an arbitrarily chosen initial approximation ${\text{|}}{{\Psi }_{0}}{{\rangle }^{{(0)}}}$

$${\text{|}}{{\Psi }_{0}}{{\rangle }^{{(k + 1)}}} = \hat {N}(1 - \alpha \hat {H}){\text{|}}{{\Psi }_{0}}{{\rangle }^{{(k)}}},$$

(A.1)

where $\hat {H} = - \frac{1}{2}\frac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}} + U$ is the Hamiltonian, $\hat {N}{\text{|}}\varphi \rangle = {\text{|}}\varphi \rangle {\text{/}}\sqrt {\langle \varphi {\text{|}}\varphi \rangle }$ is the normalization of the wavefunction, and the parameter $\alpha \ll 1$ is chosen such that the highest continuum states decay on the given grid. Since the highest continuum state corresponds to the Nyquist frequency, it suffices to require $\alpha < (2\Delta x{\text{/}}\pi {{)}^{2}}$, where $\Delta x$ is the grid step. It should be noted that a sufficiently large number of iterations, up to 100 000, are required for the ground state function ${\text{|}}{{\Psi }_{0}}\rangle$ obtained in this way to be orthogonal to the rest of the wavefunctions and not to lead to artifacts in the numerical solution of the time-dependent Schrödinger equation (2). Higher bound states ${\text{|}}{{\Psi }_{1}}\rangle$ and ${\text{|}}{{\Psi }_{2}}\rangle$ are found similarly, but under the condition of orthogonality to previously found states:

$${\text{|}}{{\Psi }_{1}}{{\rangle }^{{(k + 1)}}} = \hat {N}(1 - \;{\text{|}}{{\Psi }_{0}}\rangle \langle {{\Psi }_{0}}{\text{|}})\left( {1 - \alpha \hat {H}} \right){\text{|}}{{\Psi }_{1}}{{\rangle }^{{(k)}}},$$

(A.2)

$${\text{|}}{{\Psi }_{2}}{{\rangle }^{{(k + 1)}}} = \hat {N}\left( {1 - \sum\limits_{j = 0}^1 {\kern 1pt} {\text{|}}{{\Psi }_{j}}\rangle \langle {{\Psi }_{j}}{\text{|}}} \right)\left( {1 - \alpha \hat {H}} \right){\text{|}}{{\Psi }_{2}}{{\rangle }^{{(k)}}}.$$

(A.3)