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Physical Review A

High-order harmonic generation and parametric oscillations

Emilio Fiordilino

Published Jan 1, 1999• 15 citations
Publisher

Abstract

The differential equation of the mean electric-dipole moment induced by a laser field in a two-level atom is that of a driven parametric oscillator with time-dependent frequency. In the case of a laser field with constant intensity, a connection with the Mathieu equation is stated and it is shown from the known theory that the spectrum emitted must present only odd harmonics of the pumping field.

DOI
10.1103/physreva.59.876

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Acta Physica Polonica A•Jun 1, 2020•1 citation
The differential equation of the mean electric-dipole moment induced by a laser field in a two-level atom is that of a driven parametric oscillator with time-dependent frequency. In the case of a laser field with constant intensity, a connection with the Mathieu equation is stated and it is shown from the known theory that the spectrum emitted must present only odd harmonics of the pumping field.
We study harmonic generation by a single atom exposed to two short perpendicularly polarized laser pulses. The two perpendicular electric fields oscillate at two different frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. Hence, the resultant field has a polarization which depends on time. Since harmonics are emitted when the resultant oscillating field is linearly polarized, it is expected that a short pulse of harmonics may be emitted if the external field is linearly polarized during a short period of time. We show that, indeed, the atom may emit an ultrashort pulse of a given harmonic. This result has been obtained by time-frequency analyzing the acceleration of the induced dipole moment with a filter whose frequency bandwidth is smaller than twice the frequency of the external field. Our calculation of the dipole acceleration is based on the numerical solution of the time-dependent Schr\"odinger equation. We then address the question of how far it is possible to reduce the duration of the emitted pulse of one given harmonic by adjusting both ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ and keeping the amplitude of this pulse significant. In order to answer to this question, we used the quantum version of the two-step model [M. Lewenstein et al., Phys. Rev. A 49, 2117 (1994)].
The interaction between radiation and a two-level molecular system in the presence of permanent dipole moments is analyzed. It is found that the nonzero difference between permanent dipoles induces a polarization that oscillates at frequencies /spl omega/, 2/spl omega/, 3/spl omega/, etc., where /spl omega/ is the frequency of the electromagnetic field resonant with the two-level system. An analysis of the propagation equation of the harmonics shows that even harmonics are amplified while odd harmonics are absorbed.
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.
The phase relationship between an induced atomic dipole moment and a driving laser field is an essential aspect of high harmonic generation (HHG) in a dense medium. A ``phase-matching'' criterion must be fulfilled to obtain coherent amplification of harmonic radiation, which is critical for practical HHG. We have performed numerical calculations on model atoms to survey the intensity dependence of the phase in the case of strong one- or two-frequency laser fields. The calculations show that multiphoton resonances significantly affect the yield of harmonic generation and that in the ``plateau'' regime, the phases exhibit only weak dependence upon the intensity of the driving field. \textcopyright{}1996 The American Physical Society.
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.