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Journal of Modern Optics

Harmonic generation from ionization-damped two-level system in the framework of nonlinear Riccati-type equation

A. Plucińska, R. Parzyński

Published Mar 12, 2007• 9 citations
Publisher

Abstract

We formulate the problem of harmonic generation by a two-level system with ionization in terms of a nonlinear Riccati-type differential equation. Different simplified forms of this equation are solved analytically and the resulting photon-emission spectra are compared with those obtained by others along different lines. For the complete-form Riccati-type equation, we give an iterative code solving this equation for arbitrary laser pulses. The code is used to study the effect of ionization and turn-on of the laser pulse on the harmonic spectrum.

DOI
10.1080/09500340601022920

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We formulate the problem of harmonic generation by a two-level system with ionization in terms of a nonlinear Riccati-type differential equation. Different simplified forms of this equation are solved analytically and the resulting photon-emission spectra are compared with those obtained by others along different lines. For the complete-form Riccati-type equation, we give an iterative code solving this equation for arbitrary laser pulses. The code is used to study the effect of ionization and turn-on of the laser pulse on the harmonic spectrum.
We formulate the problem of a two-level system in a linearly polarized laser field in terms of a nonlinear Riccati-type differential equation and solve the equation analytically in time intervals much shorter than half the optical period.The analytical solutions for subsequent intervals are then stuck together in an iterative procedure to cover the whole scale time of the laser pulse.Very good quality of the iterative method is shown by recovering with it a number of subtle effects met in earlier numerically calculated photon--emission spectra from model molecular ions, double quantum wells, atoms, and semiconductors.The method is used to describe novel, carrier-envelope offset phase effects in the region of extreme nonlinear optics, i.e., when two--level systems are exposed to pulses of only a few cycles in duration and strength ensuring the Rabi frequency to approach the laser light frequence.
In the limits of weak and strong excitation, we solve analytically the Riccati-type differential equation for a two-level system in the field of two different lasers and then use these solutions to find easy to handle formulae for the induced electric-dipole moment of the system.We apply these formulae to describe the effects of strong excitation in both the higher harmonics generation and multiphoton frequency mixing, namely the diminishing of the generation of harmonics of a given laser when an additional laser is turned on and the diminishing of the radiation at mixed frequencies with increasing strength of lasers under the condition of multiphoton excitation.
Linearized version of the Riccati-type differential equation for the ratio of population amplitudes in a laserdriven two-level system is used to calculate analytically the induced electric dipole moment.The formula found for the dipole moment is valid for weak excitations by smooth-shape pulses of arbitrary off-resonant frequencies, i.e., those producing neither exact one-photon nor odd multiphoton resonances.For a given pulse shape, the formula allows to express the Fourier components of the induced dipole moment as functions of both laser frequency and intensity and to find the field-dependent refractive index of the system versus laser frequency.
We theoretically study intensity-saturation effects in resonant harmonic generation and multiphoton ionization. A first laser excites a two-photon resonance which is coupled by a second laser to an autoionizing state. Starting from an effective Hamiltonian for the three resonant atomic states, we derive a set of equations for the density matrix of the gaseous medium and the electromagnetic field within a semiclassical framework. We present and discuss analytical and numerical solutions of these equations which show a variety of line profiles depending critically on the intensities of the incident laser pulses.
In an approach based on the nonlinear Riccati-type differential equation for the ratio of the level population amplitudes, rather than on the optical Bloch equations, we describe the response of a two-level system to a few--cycle light pulse of intensity from the regime of extreme nonlinear optics, i.e., when the Rabi and transition frequencies are comparable.Emphasis is put on the dependence of the spectra of the scattered light on the carrier--envelope phase, duration and strength of both resonant and off-resonant pulses.
Laser excitation of a two-level atom is analytically studied for arbitrary pulse shape, area, and energy. The ionization mechanism is modeled as a homogeneous width $\ensuremath{\gamma}$ of the atomic excited state. A transformation of the optical Bloch equations allows the dynamics to be completely described by a single equation for the area of the pulse at time ttt, $\ensuremath{\Phi}(t)$. Analytic solutions for the total bound population for a wide variety of pulse shapes can be found in two regions: $\ensuremath{\Phi}(t)\ensuremath{\approx}sin\ensuremath{\Phi}(t)$ and $\ensuremath{\Omega}(t)>\ensuremath{\gamma}$, where $\ensuremath{\Omega}(t)$ is the resonance Rabi frequency. The generalization of the two-level atom results to the NNN-level atom is discussed.